LUiKARY    OF    THK 


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ELEMENTARY 


ALGEBRA: 


Klf  BR  A  CI  NO 


THE     FIRST    PRINCJPLES 


THE   SCIENCE 


BY   rHARLES    DAVIE  S,  LL.D., 

AUTHOR   u  r 

AIimiLrETIC,     ELKVENTART     eEOllIMItT,     ELEKENTB     OF     aOKVir. 

KI^IIKNTS   OF    DESCKIITI-V-E    AND    ASALYTIOAL   OKOMETRY^ 

USKTS   OF   UIKKEIJENTIAL   AND    INTEGRAL   CALC 

AUD    A   TREATISE    ON    PHADE8,    SHADOW^^A^' 
AND   PERSPECTIVK. 


NEAV   YORK: 
A.   S.   BARNES    &    BURR, 

51  A  ft3  JOHN  STREET. 

SOLD   BT   BOOKSBLLESa,  OEMRBALLT,  TUBOUCIIOUT  TUB  UNITED  8TATE8» 

1860. 


3>3 


A.    S.    BARNES    &    COMPANY  S    PUBLICATIONS. 
Davies'  Course  of  M  ath  em  atics. 

MATHEMATICAL    WORKS, 

IN    A.  SERIES   or  THEEE  PABT3  : 

ARITHMETICAL,  ACADEMICAL,  AND  COLLEGIATE. 
BY  CHARLES  DATIES,    LL.D. 

DAVIES'  LOGIC  AND  UTILITY   OF  MATHEMATICS. 

Tut*  series,  combining  all  that  is  most  valuable  in  the  various  methods  of  Euro- 
pean instruction,  improved  and  matured  by  the  suggestions  of  more  than  thirt]^ 
years'  experience,  now  forms  the  only  complete  consecutive  course  of  Mathematics. 
Its  m»;thods,  harmonizing  as  the  works  of  one  mind,  carry  the  student  onward  by 
the  same  analogies  and  the  same  laws  of  association,  and  are  calculated  to  impart 
a  comprehensive  knowledge  of  the  science,  combining  clearness  in  the  several 
branches,  and  unity  and  proportion  in  the  whole  ;  being  the  system  so  long  m  use 
at  West  Point,  through  which  so  many  men^  eminent  for  their  scientific  attainments, 
have  pjissed,  and  having  been  adopted,  as  X'^^*  Books,  by  most  of  the  colleges  in  th* 
Vnited  States.  5X   3*2^4^ 

I.    THE    ARITHMETICAL   COURSE  FOR    SCHOOLS 

1.  PRIMARY    ARITHMETIC    AND    TABLE-BOOK. 

2.  INTELLECTUAL    ARITHMETIC. 

S.  SCHOOL  ARITHMETIC.     (Key  Separate.) 

4.    GRAMMAR    OF    ARITHMETIC. 

II.     THE    ACADEMIC    COURSE. 

1.  THE  UNIVERSITY  ARITHMETIC.     (Key  Separate.) 

2.  PRACTICAL   MATHEMATICS    FOR    PRACTICAL    MEN. 

3.  ELEMENTARY  ALGEBRA.'     (Key  Separate.) 

4.  ELEMENTARY    GEOMETRY   AND    TRIGONOMETRY. 

5.  ELEMENTS    OF    SURVEYING. 

III.     THE    COLLEGIATE    COURSE. 

1.  DA  vies'    bourdon's    ALGEBRA. 

2.  DAVIES'    LEGENDRe's    GEOMETRY    AND    TRIGONOMETRY. 

3.  DAVIES'    ANALYTICAL    GEOMETRY. 

4.  DAVIES'    DESCRIPTIVE    GEOMETRY. 

5.  DAVIES'    SHADES,    SHADOWS,    AND    PERSPECTIVE. 

6.  DAVIES'    DIFFERENTIAL   AND    INTEGRAL    CALCULUS. 

Entered  according  to  the  Act  of  Congress,  in  the  year  one  thousand  eight  hundred 
iwd  fifty-two,  by  Charles  Daviks,  in  the  Clerk's  Ofiice  of  the  District  Court  of  the 
iJniled  States,  for  the  Southern  District  of  New  York. 

JL  P.  WNH   AMD  ea      BTKBKOTYrmS. 


PREFACE. 


Algebra  naturally  follows  Arithmetic  in 

I  course  of  scientific  studies,  yet.  ihe  change  from  the 
ethods  of  reasoning  on  numbers  to  a  system  of  rea- 
ining  entirely  conducted  by  letters  and  signs,  is  rather 
>rupt  and  not  unfrequently  discourages  the  pupil. 
In  this  work,  it  has  been  the  intention,  to  form  a 
mnecting  link  between  Arithmetic  and  Algebra,  to 
lite  and  blend,  as  far  as  possible,  the  reasoning  on 
imbers  with  the  more  abstruse  methods  of  analysis. 

The  Algebra  of  M.  Bourdon  has  been  closely  follow- 

K  Indeed,  it  has  been  a  part  of  the  plan,  to  furnish 
mtrod action  to  that  admirable  treatise,  which  is 
stly  considered,  both  in  Europe  and  this  con '^ try,  as 
the  best  work  on  the  subject  of  which  it  treats,  that 
has  yet  appeared.  The  work  of  Bourdon,  however, 
even  in  its  abridged  form,  is  too  voluminous  for  schools, 
and  the  reasoning  is  too  elaborate  and  metaphysical 
for  beginners. 

It  has  been  thought  that  a  work  wliich  should  so  far 
modify  the  system  of  M.  Bourdon  as  to  bring  it  within 
the  scope  of  our  common  schools,  by  giving  to  it  a 
more  practical  and  tangible  form,  could  not  fail  to  be 
useful.  Such  is  the  object  of  the  Elementaky 
Algebra. 


JV  PREFACE. 

The  success  whicli  has  attended  this  effort,  so  to 
simplify  the  subject  of  Algebra  as  to  bring  it  within 
the  range  of  common  school  instruction,  has  been  pecu- 
liarly gratifying.  It  is  about  twelve  years  since 
the  first  publication  of  the  Elementary  Algebra. 
Within  that  time,  between  twenty  and  thirty  editions 
have  been  printed,  and  several  works  of  other  authors, 
have  also  appeared,  modelled  after  the  same  general  plan, 

In  the  present  edition,  few  alterations  have  been 
made  in  the  general  plan  of  the  work.  The  introduc- 
tion has  been  somewhat  enlarged  for  the  purpose  of 
preparing  the  pupil  by  a  thorough  system  of  mental 
training,  for  those  processes  of  reasoning  which  are 
peculiar  to  the  algebraic  analysis. 

I  have  availed  myself,  in  the  present  edition,  of 
majiy  valuable  suggestions  from  teachers  who  have 
•tsed  the  work,  and  favored  me  with  their  opinions 
both  of  its  defects  and  merits. 

The  criticisms  of  those  engaged  in  the  daily  business 
of  teaching  are  invaluable  to  an  author;  and  I  shall 
feel  myself  under  special  obligations  to  all  such  who 
will  be  at  the  trouble  to  communicate  to  me,  at  any 
time,  such  changes,  either  in  methods  or  language,  as 
their  experience  may  point  out.  It  is  only  through 
the  cordial  co-operation  of  teachers  and  authors — by 
joint  labors  and  mutual  efforts,  that  the  text-books  of 
the  country  can  be  brought  to  any  reasonable  degree 
of  perfection. 


FisHKiLL  Landing, 
July,  1862. 


CONTENTS 


CHAPTER    I. 

PRELIMIKA&T    DEFINITIONS    AND   REXARKS. 

fcRTICLB* 

jlra — Definitions — Erplcnation  of  the  Algebraic  Signs,     -  1 — 23 

limilar  Terms — Reduction  of  Similar  Terms,          -         -         -  28 — 26 

Addition— Rule,    ...                  26—28 

Subtraction— Rule— Remark, 28—38 

Multiplication — Rule  for  Monomials, 83 — 36 

lie  for  Polynomials  and  Signs, 86 — 88 

jmarks — Properties  proved,         ..-.-.  88 — 42 

ivisiou  of  Monomials — Rule,         ...        -         -  42 — 46 

lification  of  the  Symbol  a\       •                 ....  45-  46 

the  Signs  in  Division, 46 — 47 

irisioD  of  Polynomials, 47 — 49 

CHAPTER   II. 

ALOEBHAIO     FRAOTIONSb 


litions — Entire  Quantity — Mixed  Quantity, 
Reduce  a  Fraction  to  its  Simplest  Terms,  - 
Rfeduce  a  Mixed  Quantity  to  a  Fraction,     - 
Reduce  a  Fraction  to  an  Entire  or  Mixed  Quantity, 
Reduce  Fructions  to  a  Common  Denominator,     - 
Tc  Add  Flections, 


49—62 
62 
68 
64 
66 
66 


VI 


CONTENTS. 


To  Subtract  Fractions,  - 
To  Multiply  Fractions,  - 
To  Divide  Fractions, 


ARTICLS8. 

57 
68 
69 


CHAPTER   III 


EQUATIONS   OF  THE   FIKST   DEGEEE. 


Definition  of  an  Equation — Properties  of  Equations, 
Transformation  of  Equations — First  and  Second,     - 
Resolution  of  Equations  of  the  First  Degree — Rule, 
Questions  involving  Equations  of  the  First  Degree, 
Equations  of   the    First    Degree    involving  Two   Unknown 

QuaBtities, 

Elimination — By  Addition — By  Subtraction — By  Comparison, 
Resolution  of  Q-iestions   involving  Two  or  more   Unknown 

Quantities, 


60—66 

66- 

-70 

70 

ni- 

-72 

72 

73- 

-76 

76—79 


CHAPTER    IV. 


OF   POWERS. 


Definition  of  Powers, 79 

To  raise  Monomials  to  any  Power, 80 

To  raise  Polynomials  to  any  Power, 81 

To  raise  a  Fraction  to  any  Power, 82 — 83 

Binomial  Theorem,         ...                          ...  84 — 90 

CHAPTER   V. 


Definition   of   Squares — Of    Square    Roots — And    Perfect 

Squares, 90 — 96 

Rule  for  Extracting  the  Square  Root  of  Numbers,       -        -  96 — 100 

Square  Roots  of  Fractions, 100 — 103 

Square  Roots  of  Monomials, 103 — 107 

Calculus  of  Radicals  of  the  Second  Degree,        -                 -  lo7 — 109 


OOHTSNTb. 


Vfi 


iJlTICLia. 

Addition  of  Radicals, 109 

^^  Bubtraction  of  Radicals, 110 

I^HMultiplication  of  Radical* Ill 

I^H)iyision  of  Radicals,  -                 112 

I^KExtraction  of  the  Square  Root  of  Polynomials,  -                  -  113—116 

I^P  CHAPTER    VI 

Eqoations  of  the  Second  Degree, 116 

rDefinition  and  Form  of  Equations,      ....  116 — 118 

[nc<iniplete  Equations, 118 — 122 

Coi!i|Nete  Equations, 122 

Four  Forms, •  128—121 

)lution  of  Equations  of  the  Second  Degree,  -        -        •  127 — 128 

Properties  of  the  Roots, 128—134 

CHAPTER   VII. 

)f  Progressions, 136 

jressions  by  Differences, 136 — 138 

[Last  Term, 138—140 

>um  of  the  Extremes — Sum  of  the  Series,          -        -        -  140 — 141 

le  Five  Numbers — To  Find  any  Number  of  Means,  -        -  141 — 144 

kometrical  Proportion  and  Progression,      ...        -  144 

'^arious  Kinds  of  Proportion, 144 — 166 

leometrical  Progression, 166 

st  Term— Sum  of  the  Seri'es, 167—171 

ressions  having  an  Infinite  Number  of  Terms,      -        -  171 — 172 

10  Five  Numbers — To  Find  One  Mean,    ....  172 — 178 

CHAPTER    VIII. 

Theory  of  Ix^arithms, 174 — 179 


SUGGESTIONS   TO  TEACHERS. 


1.  The  Introduction  is  designed  as  a  mental  exercise.  If 
thoroughly  taught,  it  will  train  and  prepare  the  mind  of  the 
pupil  for  those  higher  processes  of  reasoning,  which  it  is  the 
peculiar  province  of  the  algebraic  analysis  to  develop.-^* 

2.  The  statement  of  each  question  should  be  made,  and 
every  step  in  the  solution  gone  through  with,  without  the 
aid  of  a  slate  or  black-board ;  though,  perhaps  in  the  be- 
ginning, some  aid  may  be  necessary  to  those  unaccustomed 
to  such  exercises. 

3.  Great  c.are  must  be  taken  to  have  every  principle  on 
which  the  statement  depends,  carefully  analyzed  ;  and  equal 
care  is  necessary  to  have  every  step  in  the  solution  distinct^ 
ly  explained. 

4.  The  reasoning  process  embraces  the  proper  connection 
of  distinct  apprehensions,  and  the  consequences  which  follow 
from  such  a  connection.  Hence,  the  basis  of  all  reasoning 
must  lie  in  distinct  elementary  ideas. 

5.  Therefore,  to  teach  one  thing  at  a  time — to  teach  that 
thing  well — to  explain  its  connections  with  other  things,  and 
the  consequences  which  follow  from  such  connections,  would 
seem  to  embrace  the  whole  art  of  instruction. 


INTRODUCTION. 


LESSON   I. 


N  and  Charles  have  twelve  apples  betweeix  tneni, 
and  each  has  as  many  as  the  other :  how  many  has  each  1 

tif  we  suppose  the  apples  divided  into  two  equal  parts,  it 
\  plain  that  John  will  have  one  part  and  Charles  the  other: 
ence,  they  will  each  have  six  apples. 
In  Algebra,  we  often  represent  numbers  by  the  letters  of 
lie  alphabet ;  that  is,  we  take  a  letter  to  stand  for  a  num- 
er.  Thus,  let  x  stand  for  the  apples  which  John  has. 
Then,  as  Charles  has  an  equal  number,  x  will  also  stand  for 
the  apples  which  he  has.  But  together,  they  ha^e  twelve 
apples,  hence,  twice  x  must  be  equal  to  12.  This,  we  write 
thus 

a:-|-a;  =  2a-=  12; 

and  if  twice  x  is  equal  to  12,  it  follows  that  once  ar,  cr  «, 
will  be  equal  to  12  divided  by  2,  or  equal  to  6.  This,  we 
write  thus : 

When  we  write  x  by  itself,  we  mean  one  ar,  or  the  same  as 
Ix.  If  we  write  2ar,  we  mean  that  x  is  taken  twice;  if  3.r, 
that  it  is  taken  three  times,  &c. 


1.  In  the  first  question,  how  many  apples  has  each  boy?  By  whal 
arc  numbers  represented  in  Algebra  ?  If  x  stands  by  itself,  how  many 
times  X  are  expressed  ?  What  does  2x  denote  ?  What  3a:  ?  What  ix, 
Ac  If  we  have  x  -f  a",  to  how  many  times  x  is  it  equal  f  If  we  hav« 
the  value  of  2x,  bow  do  we  find  the  value  of  x  ? 

1 


A\ 


'Z  ELEMENTARY     ALGEBKA. 

2.  James  and  John  together  have  24  peaches,  and  one  has 
as  many  as  the  other  :  how  many  has  each  1 

Let  X  stand  for  the  number  of  peaches  which  James  has : 

then  X  will  also  denote  the  number  of  peaches  which  John 

has ;  and  since  they  have  24  between  them, 

x-\-  x  =  24.; 

24 
that  is,         2ic  =  24     and     x  =  —  =  12. 

Therefore,  each  has  twelve  peaches. 

*^    William  and  John  have  36  pears,  and  one  has  as  many 
as  tne  other  :  how  many  has  each  1 

Let  the  number  which  each  has  be  denoted  by  x. 
men  a;  -f-  a;  =  36 ; 

that  is,         2x  =z  36     and     a;  =  — -  =  18. 

4.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  20 1 

Let  the  number  be  denoted  by  x  :  then,  as  the  number  is 

to  be  added  to  itself,  we  have 

a;  -f  a:  =  20 ; 

20 
that  is,  2a;  =  20     or     x  =  —  =  10. 

<^ 

Hence,  10  is  the  number. 

5.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  30  1 

2>  In  the  second  question,  what  does  x  stand  for?     What  is  twice  x 
equal  to  ?     How  then  do  you  find  the  value  of  x  ? 

3.  In  tlie  third  question,  what  does  x  stand  for  ?     What  is  x  equal  to ! 
How  do  you  find  the  value  of  x  ? 

4.  In  the  fourth  question,  what  does  x  stand  for  ?     What  is  twice  a 
equal  to  ?     How  do  you  then  find  x  ? 

5.  In  the  fifth  question,  what  does  x  stand  for  ?    How  do  you  find  itsi 
value  ? 


NTR  DDUCTION. 


0    What  number  is  that  which  added  to  itself  will  give  a 
1^^^     ium  r?q'ial  to  50? 

7.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  100? 

8.  What  number  added  to  itself  will  give  a  sum  equal 
^^Bto  80 1 

^H    9.  What  number  added  to  itself  will  give  a  suna  equal 
^■fo)25? 

^^     10.  What  number  added  to  its«Jf  will  give  a  sum  equal 
to  371  ] 


1.    JOHl 


LESSON    II 


OHN  and  Charles  together  have  12  apples,  and  Charles 

^^^has  twice  as  many  as  John  :  how  many  has  eachl 

^^H     if  we  now  suppose  the  apples  to  be  divided  into  three 

^^^Bequal  parts,  it  is  fevidont  that  John  will  have  one  of  the 

^^Hparts  and  Charles  two  of  them. 

^H     Let  us  denote  by  x  the  number  of  apples  which  John  has. 

^HThen  2x  will  denote  what  Charles  has,  and  x  -\-2x  will  be 

^^|equal  to  the  whole  number  of  apples.    This  equality  is  thus 

^Hexpressed : 

^B  x-\-2x=  12* 


12 


3x  =  12     or     x  =  —  =  4, 


that  is, 

therefore,  John  has  4  apples,  and  Charles  8. 

6i  How  do  you  find  the  value  of  z  in  the  .6th  question  ?  How  in  the 
8th  t     How  in  the  9th  ?     How  in  the  10th  ? 

QrK8TioN8  ON  Lekson  II. — 1.  luto  how  many  parts  do  we  suppose 
Ihe  12  apples  to  be  divided?  How  many  of  the  parts  will  Johrj  have! 
What  is  the  value  of  each  part  ?  If  x  stands  for  one  of  the  parts,  what 
will  stand  f<tr  two  parts!  What  for  three  part«f  If  you  hapo  tLe 
VMlue  of  Sx,  bow  will  you  Hud  the  value  uf  x  t 


4  ELEMENTARY     ALGEBRA. 

2.  James  and  John  have  30  pears,  and  John  has  twice  as 
many  as  James  :  how  many  has  each  1 

Here,  again,  let  us  suppose  the  whole  number  to  be  divided 
into  three  equal  parts,  of  which  James  must  have  one  part, 
and  John  two. 

Let  us  then  denote  by   x^  the  number  of  pears  which 

James  has :  then  2.r  will  denote  the  number  of  pears  which 

John  has,  and  x  -\-'\lx  will  be  equal  to  the  whole  number  of 

pears :  and  we  shall  have 

a;  4-  2a;  =  30  ; 

30 
that  is,  3a:  =  30     or     a;  =  —  =  10. 

o 

3.  William  and  John  have  48  quills  between  them,  and 
John  has  twice  as  many  as  William :  how  many  has  eachi 

Let  the  number  of  quills  which  William  has  be  denoted 

by  X :  then,  since  John  has  twice  as  many,  his  will  be  de 

noted  by  2a;,  and  the  number  possessed  by  both,  will  be  da 

noted  by  a;  +  2a;.     Hence,  we  shall  have 

a;  -I-  2a;  =  48 ; 

48 
that  is,  3a;  =  48     or     a;  21:  —  =  16. 

o 

Hence,  William  has  16  quills,  and  John  32. 

4.  What  number  is  that  which  added  to  twice  itself,  wll^ 
give  a  sum  equal  to  60  ] 

Let  the  number  sought  be  denoted  by  a;,  then  twice  thj 
number  will  be  denoted  by  2a;,  and  we  shall  have 
a:  4-  2a;  =  60  ; 

that  Is,  3a;  =  60     or     a;  =  —  =  20 ; 

o 

und  we  see  that  20  added  to  twice  itself  will  give  60. 

"It  In  question  second,  what  is  the  value  of  one  of  the  parts  ?  Wha^ 
iu  queotion  3d  ?     How  do  you  state  question  4th  % 


?>> 


m 


/ 

HTRODUOTldTH^^^  5 

5.  John  says  to  Charles,  "give  me  your  marbles  and  I 
ffhall   have  three  times  as  many  as  I  have  now."     "  No," 

ys  Charles,  "  but  give  me  yours,  and  I  shall  have  just  51." 
How  many  had  each  ? 

Let  the  number  of  marbles  which  John  has  be  denoted 

by  z:  then,  "Zx  will  denote  the  number  which  Charles  has^ 

and  since  they  have  51  in  all,  we  write 

a;  -h  2x  =  51 ; 

51 
that  is,  3x  =  51     or     x  =  —  =  17. 

o 

6.  What  number  is  that  which  added  to  twice  itself  will 
give  a  sum  ecjual  to  75  ? 

Let  the  number  be  denoted  by  x:  then,  twice  the  number 

will  be  expressed  by  2Xj  and 

a:  -f  2ar  =  75  ; 

75 
that  is,  3a:  =  75     or    x  =  —  =  25. 

o 

7.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  90  ] 

8.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  57 1 

9.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  39  ? 

10.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  21? 


LESSON   III. 
1.  If  James  and  John  together  have  24  quills,  and  John 
has  three  times  as  many  as  James,  how  many  has  each? 

§•  How  do  you  state  question  6th  ?  Explain  the  6th  question  t  Also 
the  7th  t  What  is  the  required  number  in  the  8th  ?  What  in  the  VtL  I 
fVhat  in  the  10th  ? 


0  ELEMENTARY     ALGEBRA. 

It  is  plain  that  if  we  suppose  the  twenty-four  quills  to  be 
divided  in  four  equal  parts,  that  James  will  have  one  of  the 
parts,  and  John  three. 

Let  us  now  denote  by  x  the  number  of  quills  which  Jamea 

has :  then  Zx  will  denote  the  number  of  quills  which  John 

h-Hs,  and  we  shall  have 

a;  +  3a;  =  24  ; 

24 
fehat  is,  4.r  =  24     or     x  =  —  =  6. 

2.  What  number  is  that  which  added  to  three  times  itself 
will  give  a  sum  equal  to  48  % 

If  we  denote  the  number  by  a?,  we  shall  have 

a;  +  3a;  =  48  ; 

48 
that  IS,  4a;  =  48     or    a;  =  -—  =  12. 

4 

3.  John  and  Charles  have  60  apples  between  them,  and 
Charles  has  threw?  times  as  many  as  John :  how  many  has  each  1 

If  we  suppose  the  number  of  apples  to  be  divided  into 
four  equal  parts,  it  is  evident  that  John  will  have  one  of 
those  parts,  and  Charles  three. 

Let  X  =  the  number  which  John  has ;  then  3a;  will  stand 
for  the  number  which  Charles  has,  and  we  shall  have 
a;  -I-  3a;  =  60 ; 

that  is,  4a;  =  60     or     x  =  —  =  15. 

Bence,  John  will  have  15  and  Charles  45. 

!•  If  the  twenty-four  quills  be  divided  into  four  equal  parts,  how 
many  parts  will  John  have  ?  How  many  will  James  have?  "What  is 
«ach  part  equal  to  ? 

2i  If  three  times  a  number  be  added  to  the  number,  how  many  times 
will  the  number  be  taken  ?  If  4a;  is  equal  to  48,  what  is  the  value  of  x  \ 
Explain  the  third  question  If  4,x  is  equal  to  60,  how  do  you  find  the 
value  of  X  I 


INTRODUCTION. 


4.  What  number  is  that  which  being  added  to  thiee  tunes 
Itself  will  give  a  sum  equal  to  100  ? 

Let  the  number  be  denoted  by  x :  then 
I  ar  -f  3j;  =  100  ; 

that  is,  4a;  =  100    or    x  ■= 


=  25. 


5.  What  lumber  is  that  which  if  added  to  four  times  it 
ielf,  the  sum  will  be  equal  to  60  1 
Let  X  denote  the  number.     Then, 
a;  -(-  4a;  =  GO ; 


that 


5ar  =  60    or 


60       ,« 

a;  =  —  =  12. 
5 


6.  What  number  is  that  which  being  multiplied  by  3,  and  the 
product  added  to  twice  the  number  will  give  a  sura  equal  to  75 1 

Let  the  number  be  denoted  by  x. 
Then,     Zx  =  the  product  of  the  number  by  3 ; 
and        2a;  =  twice  the  number  ; 
and  3a;  4-  2a;  =  5a;  =  75  ; 

75 

or  a;  =  —  =  15,  the  required  number, 

o 

7.  What  number  is  that  which  being  added  to  three  times 
itself  will  give  a  sum  equal  to  140  1 

8.  What  number  is  that  which  being  multiplied  by  5,  and 
the  product  added  to  the  number,  will  give  a  sum  equal  to  240? 

9.  What  number  is  that  which  being  multiplied  by  2,  and 
then  by  3,  and  the  products  added,  will  give  125  ? 

5>  If  a  number  be  added  to  four  times  itself^  how  many  times  will  the 
number  be  taken  \ 

6.  If  X  stands  for  any  number,  what  will  stand  for  three  times  that 
number!     What  for  twice  the  number?     Explain  the  7th  question 
How  do  you  state  it  f     What  is  \x  equal  to  ?     Why  I    How  then  do 
you  find  x !    How  do  you  state  the  8th  question  ?    What  is  6a;  equal  to  I 
How  then  do  you  find  x  ? 

0.  If  a:  denotes  a  number,  what  will  otand  for  twice  the  number  I 
What  for  llirc4i  (Huca  thu  iiuiuUtI 


ELEMENTARY      ALUEUUA, 


LESSON    IV. 

1.  John  and  Charles  together  have  80  apples,  and  Chur*  *i 
has  four  times  as  many  as  John  :  how  many  has  each  1 

If  we  suppose  the  80  apples  to  be  divided  into  5  equal 
parts,  it  is  evident  that  John  will  have  one  of  the  partSj  and 
Charles  four. 

Let  X  stand  for  the  number  of  apples  which  John  has : 

then  4iX  will  stand  for  the  number  which  Charles  has  ;  and 

a:  +  4a;  =  80  ; 

80 
that  is,  hx  =  80     and    x  =  —-=\Q. 

5 

2.  What  number  added  to  four  times  itself  will  give  a  sum 
equal  to  90  1 

3.  What  number  added  to  five  times  itself  will  give  a  sum 
equal  to  120  ? 

4.  What  number  added  to  six  times  itself  will  give  a  sum 
equal  to  245 1 

5.  What  number  added  to  seven  times  itself  will  give  a 
sum  equal  to  360 1 

6.  What  number  added  to  five  times  itself  will  give  a  sum 
equal  to  200  1 

7.  What  number  added  to  itself  and  the  sum  to  four 
times  the  number  will  give  a  sum  equal  to  72  1 

\,  li  X  stands  for  John's  apples,  what  will  denote  Charles'  ?  What 
will  stand  for  the  apples  which  they  both  have  ?  If  bx  is  equal  to  80, 
what  will  X  be  equal  to  ?  If  a  number  be  added  to  four  times  itself. 
bow  many  times  will  the  number  be  taken  ?  If  5  times  a  number  i» 
equal  to  90,  what  is  the  value  of  the  number  ?  Explain  example  3d 
Explain  question  4th.  What  does  x  stand  for  ?  Explain  the  6th  quea 
iiua     Explain  example  Cth. 


INTRODUCTION.  V 

LESSON   V. 
1     What  number  added  to  five  times  itself,  will  give  a 
wum  nqual  to  60 1 

2.  John  has  a  number  of  marbles  and  buys  four  times  as 
many  more,  when  he  has  seventy -five :  how  many  had  he 
Rt  first  1 

3.  If  X  be  taken  seven  times,  and  then  eight  times,  how 
many  times  will  it  be  taken  in  all  1 

4.  If  X  be  made  equal  to  5,  in  the  last  example,  what  will 
be  the  numerical  value  of  the  sum  ? 

5.  Find  two  numbers  whose  sums  shall  be  fifty,  and  one 
of  them  four  times  the  other  ? 

Let  X  denote  the  less  number : 
then,  4x  will  denote  the  greater : 
and  by  the  conditions  of  the  question 
x  +  4x=50', 

50 
hence,  5a:  =  50  ;     or     a;  =  —  =  10. 

o 

6.  Find  two  numbers  whose  sum  shall  be  forty-five,  and 
one  of  them  eight  times  the  other. 

'  7.  Divide  the  number  thirty  into  two  such  parts  that  the 
greater  shall  be  four  times  the  less. 

8.  Divide  the  number  forty-eight  into  two  such  parts  that 
the  greater  shall  be  five  times  the  less. 

9.  Divide  the  number  sixty-four  into  two  such  parts  that 
the  greater  shall  be  seven  times  the  less. 

10.  What  is  the  sum  of  9x  and  three  ar?  What  is  the 
sum  equal  to,  numerically,  if  ar  is  equal  to  5 1 

1 1.  What  is  the  sum  of  eight  x  and  one  x  1  What  is  th«» 
sum  equal  to.  numerically,  when  a:  is  7  ? 

1* 


10  ELEMENTARY     ALGEBHA. 

12.  What  is  the  sum  of  x  +  x -\- Sx  +  4x  A- 5x1  What 
is  this  sum  equal  to,  numerically,  if  a;  is  2  ? 

13.  What  is  the  sum  of  2x  +  x  -{-  Sx  -\-  4x  4-  x  -^  xl 
What  is  the  sum  equal  to,  numerically,  when  a;  is  9  1 

14.  James  and  John  wish  to  share  thirty-six  apples,  so 
that  James  shall  have  three  times  as  many  as  John  :  how 
many  will  each  have  ? 

15.  What  number  added  to  eight  times  itself,  and  this 
Bum  to  three  times  the  number,  will  give  a  sum  equal  to  48 1 

16.  What  number  is  that  whose  ninth  part  added  to  the 
number  will  give  a  sum  equal  to  twenty  ? 

Let  the  number  be  denoted  by  9x : 

then  one-ninth  of  9a;  will  be  denoted  by  x  ;  and  by  the  con 

ditions  of  the  question 

9a;  +  a;  =  20, 

20 
hence,  10a;  =  20     or     a;  =  —  =  2. 

Then,  if  a;  =  2,    9a;  =  18,  the  number  sought. 


'      LESSON    VI. 

1.  What  number  added  to  six  times  itself,  and  then  to 
five  times  itself,  will  give  a  sum  equal  to  twenty -four? 

Let  X  denote  the  number  : 
then,  6x  =     six  times  the  number, 

and  5x  =     five  times  the  number . 

and  by  the  conditions  of  the  question 

a;  +  6a;  +  5a;  =  24  : 

24 
hence,  •12a;  =  24     or    x  —  —  =  2. 


INTRODUCTION.  I 

2  What  number  added  to  twice  itself,  then  to  three  time^^ 
[itself,  to  four  times  itself,  and  to  five  times  itself,  will  give 
&  sum  equal  to  fifteen  ? 

3.  Divide  twenty-one  into  three  such  parts,  that  the  second 
shall  be  equal  to  four  times  the  first,  and  the  third  to  four 
times  the  second. 

4.  A  farmer  has  three  times  as  many  sheep  as  goats,  and 
one-third  as  many  lambs  as  sheep :  he  has  thirty  in  all : 
how  many  has  he  of  each  sort  ? 

Let  X     denote  the  number  of  goats  : 

tnen  3a;     will  denote  the  number  of  sheep, 

and  X    will  denote  the  number  of  lambs ; 

and  by  the  conditions  of  the  question 

X  -\- Sx -{•  X  =  oO     the  number  in  all. 

30 
Then  5ar  =  30 ;  or  2:  =  —  =  6,     the  lambs  or  goats. 

Also,  3a;  =  3x6  =  18    the  number  of  sheep. 

5.  James  has  twice  as  many  dime-pieces  as  cent-pieces, 
and  has  forty-two  cents  in  all :  how  many  dime-pieces  has 
hel 

6.  John  has  two  sisters  and  one  brother,  and  wishes  to 
divide  thirty  dollars  between  them.  He  wishes  to  give  the 
elder  sister  twice  as  much  as  the  younger,  and  the  brother 
as  much  as  both  the  sisters  :  how  much  must  he  give  to 
each? 

7.  An  orchard  contains  thirty-five  trees.  There  is  an 
equal  number  of  plum  trees  and  pear  trees ;  but  there  are 
three  times  as  many  cherry  trees  as  plum  trees,  and  twice 
as  many  apple  tree*  as  pears  :  how  many  of  each  .sort  1 


10  ELEMENTARY     ALGEBRA. 

8.  Divide  twenty-four  into  three  such  parts  that  the 
second  shall  be  double  the  first,  and  the  third  three  times 
the  first. 

9.  Divide  the  number  fourteen  into  three  such  parts,  that 
the  second  shall  be  double  the  first,  and  the  third  double 
the  second. 

10.  John  has  three  times  as  many  marbles  as  William 
has  tops :  the  tops  cost  three  cents  a  piece,  and  the  marbles 
one  cent,  and  together  they  cost  thirty  cents :  how  many 
had  each? 

11.  Jane  has  a  blush  rose  bush,  a  moss  rose  bush,  and  a 
white  rose  bush  ;  and  together  they  have  thirty-three  buds ; 
the  buds  on  the  second  are  double  those  on  the  first,  and 
those  on  the  third  four  times  those  on  the  second:  how 
many  on  each? 


LESSON    VII. 

1.  Jambs  has  three  times  as  many  marbles  as  Charles, 
and  together  they  have  thirty-two :  how  many  has  each  1 

Let     X  =  the  number  of  marbles  which  Charles  has : 
then      3a;  =  the  number  James  has  : 
and         X  -{-  Sx  z=  S2  what  both  have. 

32 

Then  4a;  =  32  ;    or    x  = —- =  S  : 

therefore,  Charles  has  8,  and  James  8x3=  24. 

2.  John,  Charles,  and  "William  have  ninety  books  :  Charles 
has  five  times  as  many  as  John,  and  William  four  times  as 
many  as  Jolm  ;  how  many  has  each  ? 


^'^OR^"li 


INTRODUCTION.  IS 

Let     x=  the  number  which  John  has: 
then     bx  =  the  number  Charles  has, 
and       4a:  =  the  number  William  has. 
Then,     3:  -f  5a;  +  4a;  =  90,  the  number  they  all  have. 

90 
Hence,  1  Ox  =  90  ;    or    a;  =  —  =  9. 

Therefore,  John  has  9 ;  Charles  45,  and  William  36. 

3.  The  sum  of  three  numbers  is  twenty -four :  the  second 
is  twice  the  first,  and  the  Ihird  five  times  the  first :  what 
are  the  numbers  '\ 

4.  The  sum  of  three  numbers  is  thirty-eight :  the  second 
is  three  times  the  first,  and  the  third  five  times  the  second : 
what  are  the  numbers  1 

5.  The  sum  of  three  numbers  is  forty -eight :  the  second  is 
seven  limes  the  first,  and  the  third  is  equal  to  the  sum  of 
the  first  and  second  :  what  are  the  numbers  1 

6.  The  sum  of  four  numbers  is  seventy  :  the  second  is 
four  times  the  first;  the  third  three  times  the  first,  and  the 
fourth  double  the  third :  what  are  the  numbers'? 

7.  Divide  the  number  thirty-nine  into  three  such  parts, 
that  the  second  shall  be  three  times  the  first,  and  the  third 
three  times  the  second. 

8.  Divide  the  number  seventy-five  into  two  such  parts, 
that  the  less  shall  be  one -fourth  of  the  greater. 

9.  Divide  one  hundred  and  thirty-three  into  three  such 
parts,  that  the  second  shall  be  three  times  the  first,  and  the 
third  five  times  the  second. 

10.  Divide  eighty-five  into  four  such  patts  that  the  second 
shall  be  four  times  the  first,  the  third  four  times  the  second 
and  the  fourth  four  times  the  third. 


14  ELEMENTARY     ALGEBKA, 


LESSON     VIII. 

WoTE. — Let  the  pupil  now  read  Articles  60,  61,  and  that  part  of  Art 
64,  which  is  found  on  page  90.     Also,  Art.  65. 

1.  James  receives  five  apples  from  John,  and  then  has 
tM  elve  :  how  many  had  he  at  first  1 

het     X  =z  the  number  he  had  at  first. 
Then       a;  +  5  =  12,  what  he  had  afterwards. 

Now,  if  X  increased  by  5,  equals  12,  x  must  be  Icsa 
than  12,  by  5.     Hence, 

a;  =  12  — 5  =  7. 
When  we  take  a  number  from  one  member  of  an  equa^ 
tion  and  place  it  in  the  other,  we  are  said  to  trans'pose  it. 

2.  William  has  eight  marbles  more  than  John,  aid  to- 
gether they  have  thirty-six :  how  many  has  eachi 

Let  X  =         the  number  which  John  has : 

then     a;  +  8  =         the  number  William  has, 
and    2a;  +  8  =  36,  the  number  they  both  have. 

Now,  if  2a;  increased  by  8  is  equal  to  36,  2a  must  be 
equal  to  36  diminished  by  8 :   hence, 

2a;  =  36  -  8  =  28 

or  a;  =  —  =  14. 

Hence,  we  see  that  a  plus  number  may  be  transposed  from 
one  member  of  an  equation  to  the  other,  by  simply  chang 
ing  its  sign  to  minus. 

3.  A  father's  age  is  double  his  son's,  and  the  sum  of 
their  ages  increased  by  four  is  equal  to  64 :  what  is  the  jig-e 
of  each  ? 


INTKODUCTIUV.  Ii^ 

Let        X  denote  the  son's  age  : 
then         2x  will  denote  the  father's  age, 
and  2x  -{-  X  will  denote  the  sura  of  their  ages. 

But  by  the  conditions  of  the  question 
2x-|-a;-(-4  =  64; 
hence,  3x  +  4  =  64 

and  3.C  =  G4  —  4  =  60, 

or  a:  =  — -  =  20,  the  son's  age. 

o 

and  20  X  2  =  40,  the  father's  age. 

4.  A.  farmer  has  three  pastures  for  sheep.  In  the  second 
he  has  twice  as  many  as  in  the  first,  and  in  the  third  as 
many  as  in  the  first  and  second  less  15,  and  he  has  in  all 
fifty -seven  :  how  many  has  he  in  each  pasture  1 

5.  What  number  is  that  to  which  if  ten  be  added,  the 
sum  will  be  equal  to  three  times  the  number  1 

6.  John  bought  an  equal  number  of  pears,  peaches,  and 
oranges ;  for  which  he  paid  one  dollar ;  he  paid  a  cent  a  piece 
for  the  pears  and  peaches,  and  three  cents  a  piece  for  the 
oranges :  how  many  did  he  buy  of  each  sort  1 

7.  A  man  deposited  in  a  savings  bank,  at  different  timea, 
eighty  dollars.  The  second  deposit  was  double  the  first,  and 
the  third  was  equal  to  the  first  and  second  and  eight  dollars 
over:  what  was  the  sum  deposited  at  each  time? 

b.  A  horse,  cart,  and  harness,  together  cost  one  hundred 
and  twenty  dollars  :  the  cost  of  the  horse  plus  twenty  dol- 
lars was  equal  to  the  cost  of  the  cart  and  harness,  and  thi» 
Q»jrt  oost  twenty  dollars  more  than  the  harness. 


16  KLEMENTAKY     ALGEBKA 

LESSON    IX. 

1.  Divide  twenty-one  dollars  between  James,  Jchn,  and 
Charles,  so  that  James  shall  have  four  dollars  more  thac 
John,  and  John  one  dollar  more  than  Charles. 

Let  X  =  James'  share  of  the  $21 : 

then         ar  —  4  =  John's  share, 
and  X  —  4:  —  \  =   Charles'  share, 
and  their  sum.  a;-fa;-f-ar  —  4  —  4  —  1  =21: 
hence,  Sx  —  9  =  21. 

Now,  if  Sx  diminished  by  9  equals  21,  3a;  must  be  equal 
to  21  increased  by  9, 

therefore,  3a;  =  21  +  9  =  30 

or,  a;  =—  =  10. 

Hence,  John's  share  =10  —  4  =  6, 
and  Charles'  share     =10  —  5  =  5. 

Remark. — We  see  from  the  above  example,  that  a  nega- 
tive number  may  be  transposed  from  one  member  of  an 
equation  to  the  other  by  simply  changing;  its  sign  to  plus. 

2.  A  person  goes  to  a  tavern  where  he  spends  three  shil- 
lings :  he  then  goes  to  a  second  and  spends  nine  shill'ngs, 
which  is  three  times  as  much  as  he  had  left :  vv^hat  had  he  at 
first? 

3.  Three  persons,  A,  B,  and  C,  spend  at  a  tavern,  twenty 
eight  dollars :  B  spends  three  dollars  more  than  A,  and  C 
seven  dollars  more  than  B  :  how  much  does  each  spend  ? 

4.  There  are  four  numbers  whose  sum  is  33  :  the  second 
is  double  the  first ;  the  third  is  three  times  the  second,  and 
the  fourth  is  four  times  the  third :  what  are  the  numbers  ? 


INTRODUCTIOir> 


'fe'i^'ORU'i 


ritjf 


17 


I 


5.  The  sum  of  two  members  is  13.  and  their  (iiff«jpeii;oe 
iiree  :  what  are  the  numbers! 

Let         X  =  the  gn^iter : 
ihen    r  —  3  =   the  less  ; 
Uid   2x  —  3  =  13  :  hence     2ar  =  13  -|-  3  =  16, 


1  /» 

— -  =  8  ;    and    8  —  ar  =  5, 


lence,  the  numbers  are  8  and  5. 

6.  James  says  to  John,  "give  me  five  of  your  marbles 
and  I  shall  have  twice  as  many  as  you  now  have"  :  together 

ey  have  nineteen  :  how  many  has  each? 

Let  X  denote  the  number  which  James  has. 
Then,  19  —  a:     will  denote  what  John  has; 

and  by  the  conditions  of  the  question, 
[  a:  +  5=2{19-ar)=38-2y 

then,  by  transposing  22?  and  5,  we  have 
8a;  =  38  -  5  =  33, 


33       1, 


I 

^H    Hemarc. — When  we  wish  to  multiply  an  algebraic  ex« 
^^)rossion,  composed  of  two  or  more  terms,  by  any  number, 

vie  place  those  terms  within  a  parenthesis,  and  wnte  the 

oiultiplier  on  the  left,  or  right.     Thus, 

2(19-ar)     or     (19  -  r)2 

denotes  that  the  difference  between  19  and  ar  is  to  be  multi 
plied  by  2. 

^K    7.  The  sum  of  the  ages  of  a  mother  and  daughter  is  50 : 
^"  the  daughter's  age  Is  one-third  of  the  mother's  age :  what  is 
the  age  of  each  ? 


18  ELEMENTAKY      ALOKBRA 


LESSON      X. 


1.  If  from  3j*  we  take  x,  what  will  remain  ?  If  we  tfikt 
away  2ar,  what  will  be  left?  H  we  take  away  3j:,  what  'aUI 
be  left  ? 

2.  If  from  Zx^  we  subtract  a;  —  1,  what  will  be  left  1 
Here  we  propose  to  take  from  Zx  a  number  less  than  2 

by  1.  If  then,  we  subtract  x  from  3ar,  leaving  2a:,  we  shall 
have  taken  too  much^  and  consequently,  the  reuiainder  w'll 
be  too  small  by  1.  Hence,  to  obtain  the  true  jemaindei 
we  must  add  1  ;  and  we  then  have 

ZX  —  {X  —  \)  :=L^X  —  X  \\  -"Ix  ^-  \. 
This,  and   all   similar  results,   are   obtained   by   merelj 
changing  the  signs  of  the  subtrahend,  and  adding  the  terms. 

3.  What  is  the  difference  between 

4^  -H  3    and    '2x  —  2 
4ar  +  3  -  (2a;  —  2)  =  4x  -  2^  -I-  3  +  2  =  2ar  -h  5. 

4.  What  is  the  difference  between 

6a:  —  9     and     2a;  —  8. 

5.  What  is  the  difference  between 

3^  —  4     and     —  x  -\-  Q, 

6.  What  is  the  difference  between 

—  5a;  -f-  7     and     -^  3a;  +  8. 

7.  James  is  three  years  older  than  John;  and  cne-si:ctl. 
of  James'  age  is  equal  to  one-fifth  of  John's, 

Let  x  denote  James'  age  ; 
then     ir  —  3  will  denote  John's ; 
and  by  the  conditions  of  the  question, 
a;       a;  —  3 
¥  ""  "~5~  ' 
hcnoc,  b'X-zzQtx  —  18,    or    x  —  18. 


IMTRODUCTION.  19 

8.  William  has  two  cents  more  than  John.  If  John*8 
^rnts  be  suhiracted  from  twice  William's,  the  remain- 
der will  be  ten  :  how  many  has  each  ? 

Lei  X  denote  the  number  of  William's; 
then      «  —  2  will  denote  John's ; 
und  by  the  conditions  of  the  equation, 

22r-(ar-2)=  10; 
that  is,        2x  —   ar-l-  2   =  10 

ar-l-  2   =  10,  or    aj  =  10  -  2  =  8. 

9.  A  farmer  has  sheep  in  two  lots.  In  one  lot  he  has  five 
more  than  in  the  other.  But  three  times  the  larger  flock  is 
equal  to  four  times  the  less :  how  many  are  there  in  each  flock  1 

10.  Lucy  is  five  years  older  than  Jane;  but  four  times 
Lucy's  age,  diminished  by  five  times  Jane  3,  is  equal  6» 
nothing  :  what  is  the  age  of  each  1 

11.  What  is  the  dilference  between 

5j;  -f-  3     and    —  7x  —  4 

12.  What  is  the  difference  between 

—  (5j:  -j-  3     and         8a:  -|-  I 


LESSON    XI. 

1.  A  grocer  buys  an  equal  number  of  lemons  and  oranges 

for  the  lemons  he  paid  two  cents  a  piece,  and  for  the  oraiig<'S 

he  paid  three  cents  a  piece,  and  for  the  whole  he  paid  eighty 

cents  :  how  many  did  he  buy  of  each  sort  ] 

Let  X  denote  the  number  of  each  kind  :   then 

2ar  =  the  cost  of  the  lemons, 

and  3ar  =  the  cost  J  the  oranges, 

and  by  the  c»">nditions  of  the  question, 

2x  +  Zx  =  80  cents. 

80 
Heiioc  Ox  =  80 ;    or    a;  =  —  =  10. 


20 


ELEMENTARY     ALGEBRA 


2.  A  grocer  buys  a  certain  number  of  lemons  at  two  cents 
A  piece,  and  tliree  times  as  many  oranges  at  four  cents  a 
piece,  and  pays  for  the  whole  eighty -four  cents  :  how  many 
does  he  buy  of  each  soil.? 

3.  What  number  is  that  which  being  added  to  five  times 
itself,  and  nine  subtracted  from  the  sum,  will  leave  a  re- 
mainder equal  to  21  1 

4.  John  has  in  his  purse  a  certain  number  of  cents,  half 
as  many  dimes  as  cents,  and  half  as  many  dollars  as  dimes  ;— 
in  all  twenty-eight  pieces  :  how  many  has  he  of  each  sort  ? 

Let  X  denote  the  number  of  dollars ; 

then  2x  will  denote  the  number  of  diiaes  ; 

and  4a:  the  cents. 

Then,  by  the  conditions  of  the  question, 

a: -f- 2a:  +  4:r  =  7a:  =  28,    or    a*  =  —  =  4. 

5.  In  a  fruit  basket  there  are  three  times  as  many  apples 
as  pears,  and  five  times  as  many  peaches  as  apples  :  in  all, 
ninety-five  :  how  many  of  each  sort  '^ 

6.  A  farmer  has  sixty-nine  head  of  eattle.  The  number 
of  cows  is  double  that  of  his  calves,  and  the  number  of 
young  cattle  is  six  times  as  great  as  his  calves;  besides,  he 
has  six  oxen  :  how  many  calves,  how  many  cows,  and  how 
uiHuy  young  cattle  1 

7.  What  number  is  that  which  being  multiplied  by  seven, 
and  five  subtracted  from  the  product,  will  give  a  result  equal 
tc  four  times  the  number  increased  by  thirteen  ? 

8.  A  merchant  has  forty-four  dollars  in  bank  bills,  in  an 
tXiual  number  of  ones,  twos,  throes  and  fives  :  how  many 
has  *ie  of  each  sort  ? 


I 


INTRODUCTION.  2) 

9.  A  jockey  has  a  horse  and  two  saddles,  one  worth 
thirty  dollars  and  the  other  five.  If  he  puts  the  best  saddle 
on  the  horse,  their  value  becomes  double  that  of  the  horse 
diminished  by  twice  the  value  of  the  other  saddle :  what  ii 
the  value  of  the  horse  1 


LESSON    XII. 

1.  What  number  is  that  to  which  if  five  be  added,  an 
the  sum  multiplied   by   three,  will  give  a  result  equal  to 
ten  times  the  numijer  plus  one? 

Let  z  denote  the  number. 
Then  by  the  conditions  of  the  question 

3(^4-5)  =  10i:-h  1, 
hence,  Sx  -{-  15  =  \0x  -{-  I  -, 

and  by  transposing  IOj:  and  15, 

^x  -\0x  =  \  -  15 
or  —lx=—  14, 

and  changing  the  signs  of  both  members, 

14 

7ar  =  14    or    x  =  —  =  2. 

Remark. — When,  after  having  brought  all  the  x's  to  the 
first  member,  the  final  sign  is  minus,  make  it  plus  by  chang- 
ing the  signs  of  all  the  terms  in  both  the  members. 

2.  The  diflTerence  of  two  numbers  is  three,  and  their  sum 
five  times  the  dificrence :  what  are  the  numbers? 

3.  James  says  to  John,  "  Give  me  your  apples,  and  I  shall 
then  have  thiee  times  as  many  as  you  have  now."  John 
says  "No;  for  the  number  of  your  apples  now  exceed* 
mine  by  four :"  how  many  had  each  1 

Let  X  denote  the  number  which  John  had. 

llien,  r  4-  4  will  denote  what  James  had  ; 


22  ELEMENTARY     ALGEBRA. 

and  by  the  conditions  of  the  question 

X  -\-  X  -j-  4  —  'Sx  ;     whence  by- 
transposing,  X  =z  4l. 

4.  James  met  some  beggars,  to  each  of  whom  he  gave  n 
cents :  had  there  been  four  more,  and  had  he  given  the  same 
to  each,  he  would  hare  given  seventy-two  cents :  how  many 
beggars  were  there  1 

5.  John  has  twice  as  many  turkeys  as  ducks;  twice  aa 
many  ducks  as  geese,  and  eight  times  as  many  chickens  ae 
geese ;  in  all,  forty-five  :  how  many  has  he  of  each  sort  1 

6.  Three  persons  receive  forty-eight  dollars  ;  the  second, 
tour  dollars  more  than  the  first,  and  the  third,  four  more 
than  the  second  :  how  much  did  each  receive  1 

7.  The  sum  of  three  numbers  is  thirty-six  ;  the  second 
exceeds  the  first  by  eight,  and  the  third  is  less  than  the 
second  by  16  :  what  are  the  numbers  ? 

Let  X  denote  the  first  /lumber ; 

then,     a;  4-  8  will  denote  the  second  ; 
and  since  the  third  is  less  than  the  second  by  16, 

a;  -f-  8  —  16  =  a;  —  8  =    the  third. 
Then  by  the  conditions  of  the  question 

x-\-x-\-S-\-x  —  S  =  SQ, 
or  3x  =  36,    or   a;  =  12. 

Hence,  the  numbers  are  12,  20,  and  4. 

Remark. — When  a  number,  as  -|-  8  and  —  8,  is  found 
twice  in  the  same  member  of  the  equation  and  with  different 
signs,  it  may  be  omitted :  and  the  two  numbers  are  then 
said  to  cancel  each  other.  If  the  same  number  is  f  )und  in 
different  members  of  the  equation  with  the  same  sign^  it 
may  be  omitted :  and  the  two  numbers  are  then  said  to 
^xincel  each  jther. 


IN  TRODUCTION. 


23 


8.  A  father,  son,  and  daughter,  on  comparing  ages,  find 
at  the  son's  age  is  dtiuble  the  daughter's:  that  twice  the 
n's  agi^  dimini.shed   by  four,  is  equal  to  the  father's   age: 

and  that  the  sum  of  their  ages  is  c^^aal  to  73 :  what  \a 

the  ago  of  eachi 


ai 

I 


LESSON   XIII. 

The  sum  of  two  numbers  is  nine  ;  if  to  the  first  six  be 
added,  the  sum  will  be  double  the  second:  what  are  the 
numbers] 

Let    r   denote  the  first  number, 

then,      9  —  X    will  denote  the  second; 

and  b)-  the  conditions  of  the  question 

X  +  6  =  2(1) -j)  =  18-2ar; 

12 
hence,  oa:  =  12 ;    or  a:  =  -^  =  4,    the  first, 

o 

and  9  —  a:  =  9  —  4  =  5,    the  second  number. 

2.  John  and  James  play  at  marbles :  James,  at  the  be- 
ginning, has  twice  as  many  as  John,  but  John  wins  eight, 
and  he  then  has  twice  as  many  as  James  has  lefl :  how 
many  had  each,  at  the  beginning] 

Let   X   denote  John's  marbles, 
then      2a?   will  denote  James' ; 
and        ar  H-  8    wh«t  James  had  after  he  won, 
ai:d        2ar— 8    denote  what  John  had  after  he  lost. 

Tnen,  by  the  conditions  of  the  question, 

ar  -f  8  =  2  (2ar-8)  =  4ar  -  16, 
cr  Sx  =  24 : 

whence,  a:  =  8,    the  number  John  had ; 

anu  2c  =  1(3,   the  number  James  hod. 


24  ELEMENTARY     ALGEBRA. 

3.  In  an  orchard  containing  sixty  trees,  theie  are  twice  as 
many  pear  trees  as  apple  trees,  and  as  many  plum  trees  as 
pear  trees  and  apple  trees  together:  how  many  trees  are 
there  of  each  sort  1 

4.  A  and  B  set  out,  at  the  same  time,  from  two  place* 
which  are  ninety  miles  apart,  and  travel  towards  each  other  ; 
A  travels  six  miles  an  hour,  and  B  three  miles  an  hour :  iu 
how  many  hours  will  they  meet  1 

Let   X  denote  the  number  of  hours : 
then       Qx    will  denote  the  number  of  miles  A  travels, 
and       Sx   the  number  of  miles  B  travels  : 
By  the  conditions  of  the  question 

Qx-\-Sx  =  90, 
whence  9a:  =  90    or   x  =  10. 

5.  Charles  buys  six  yards  of  cloth  at  a  certain  price,  and 
afterwards  nine  yards  more  at  the  same  price,  but  the  last 
time  he  paid  twenty-seven  shillings  more  than  before:  how 
much  did  he  pay  a  yard  1 

6.  A  cask  which  holds  eighty  gallons  is  filled  with  a  mix- 
ture of  brandy,  wine,  and  cider :  there  are  ten  gallons  more 
of  cider  than  of  wine,  and  as  much  brandy  as  of  cider  and 
wine  together :    how  many  gallons  are  there  of  each  1 

7.  Four  men  build  a  boat  together,  which  cost  one  hun- 
dred and  twenty-one  dollars  :  the  second  paid  twice  as  mu3h 
as  the  first,  the  third  as  much  as  the  first  and  second  together, 
and  the  fourth  as  much  as  the  third  and  second  togethei' : 
what  did  each  pay? 

8.  B  has  six  shillings  more  than  A ;  C  has  six  shillings 
more  than  B  ;  D  has  six  shillings  more  than  C ;  D  has  al  x) 
three  times  as  many  shillings  as  A  :  how  many  shillings  1  is 
eacli  ] 


INTnODDCTIOIf 


25 


At  an  election  one  hundred  votes  were  given,  and  the 
ssful  candidate  had  a  majority  of  twenty  :  how  many 

es  had  each  candidate  1 

Two  men  together  had    twenty  dollars;    and   they 

yed  till  one  lost  five  dollars,  when  the  winner  had  four 

es   as    much  as  the   loser:    how  much  had  each  when 

y  began  1 

11.  Divide  fifteen  into  two  parts,  such  that  one  part  shall 
equal  to  twice  the  other. 

12.  A  fish  was  caught  which  weighed  twenty  pounds;  the 
d  weighed  four  times  as  much  as  the  tail,  and  the  body 
ighed  five  times  as  much  as  the  tail]  What  did  each 
't  weigh"? 


LESSON    XIV 


John  has  a  certain  number  of  marbles:  Charles  haa 
half  as  many,  and  James  ono-third  as  many  :  togethei  chey 
have  eleven  :  how  many  has  each  ? 

Let    X   =    the  number  of  John's  marbles ; 

X 


then, 


.    =    the  number  which  Charles  has, 


and 


—  =    the  number  which  James  has. 


Then,  by  the  conditions  of  the  question, 

X  X 


To  clear  the  equation  of  fractions,  multiply  each  mernbef 
by  the  least    common    multiple  of  the  denominatoriJ  (j»e» 
Art.  68),  which  in  this  case  is  G,  and  we  shall  have 
2 


26  ELEMENTARY     ALGEBRA. 

6.C  -\-Sx-^2z  =  66, 
heiioe,  11a;  =  66    or   x  =  — =6. 

2.  What  is  the  sum  of  — ,      -—      and      —  ! 

2         3  4 

SOLUTION 

2   ^3    ^4    ~  12  "^    12  "^  12  ~  12  ~  ^^' 
rience,  the  sum  is  1|^2^* 

3.  What  number  is  that,  which  being  added  to  half  itself, 
to  one-third  of  itself,  and  to  one-fourth  of  itself,  will  give  a 
sum  equal  to  twenty -five  1 

4.  What  number  added  to  its  fifth  part,  will  give  a  result 
equal  to  twice  the  number  diminished  by  eight? 

5.  What  number  is  that  to  which,  if  three-fourths  of  it 
self  be  added,  the  sum  will  be  twice  the  number  diminished 
by  two? 

6.  A  farmer  has  twice  as  many  oxen  as  horses ;  and  one- 
third  the  number  of  his  horses,  added  to  half  the  number 
of  his  oxen,  is  equal  to  four:  how  many  oxen  and  horses 
had  l.e  ? 

7.  James  has  fifteen  oranges,  which  are  three-fourths  as 
many,  as  John  has,  less  three  :  how  many  has  John  ? 

8.  The  smaller  of  two  numbers  is  five-eighths  of  the 
larger,  and  their  sum  is  sixty-five  :  what  are  the  numbers  1 

9.  ITie  smaller  of  two  numbers  is  three-fourths  the 
larger,  and  their  difference  is  equal  to  half  the  greater 
liminished  by  two:  what  are  the  numbers? 

10.  A  farmer  sold  a  cow  and  calf:  he  received  one-fifth 
as  much  for  the  calf  as  for  the  cow ;  and  the  difference  be- 


INTKODt  CTIOK. 


27 


i 


i 


Iween  the  two  sums  was  twenty  four  dollars:  what  did  hb 

receive,  for  each  1 

11.  James  is  six  years  older  than  John;  and  the  sum  of 
eir  ages  plus  one-fourth  of  John's  age,  is  equal  to  tw*uty- 
ir :  what  is  the  age  of  each  ? 

1*2.  Nancy's  age  is  three  times  Eliza's:  one-half  Nanviy*8 
lus  one-third  of  Eliza's,  is  equal  to  the  ditlerence  of  theif 
38  diminished  by  one  :   what  is  the  age  of  each? 

13.  John  is  nine  years  older  than  his  sister.  If  one  sixth 
of  his  age  be  added  to  his  sister's,  the  sum  will  be  two-thirda 
of  J  ohn's.     W  hat  was  the  age  of  each  ? 

14.  The  difference  between  two  numbers  is  four;  and 
one-third  of  the  less  plus  one-fourth  the  gi  eater  is  equal  to 
6alf  the  greater :  what  are  the  numbers? 

15.  A  pr)le  is  one-third  in  the  mud,  one-half  in  the  water, 
d  six  feet  out  of  water  :   w  hat  is  the  length  of  the  pole  1 

IG.  The  weight  of  a  fish  is  thirty-two  pounds:  one-third 
the  weight  of  his  head  is  eq\ial  to  the  weight  of  his  tail; 
and  the  weight  of  his  body  is  four  times  the  weight  of  his 
tail:  what   is   the   weight  of  each  part  1 

17.  A  person  in  play  lost  one-f<«urth  of  his  money,  when 
he  found  that  he  had  one-half  of  what  he  began  with  and 
five  shillings  over:  how  much  had  he  when  he  began  to 
play? 


LESSON   XV. 


1.  The  sum  of  the  ages  of  Jane  and  Catharine  is  elgli- 
Amu :  but  one-half  of  Catharine's  age  is  equal  to  onefourtli 
uf  Joiio's  age :  what  is  the  age  of  c»ch  1 


28  ELEMENTARY     ALGEBRA.. 

Let   X    denote  Jane's  age ; 
then      18  — x   will  denote  Catharine's  age; 
and  hy  the  conditions  of  the  question 
18  —  a;  _  X 

Multiplying  each  member  of  the  equation  by  four,  we 

have 

36  —  2x  z=ix;    whence   36  =  Zx 

36 
or  3a:  =  36,    or   x  =  —  =  12. 

o 

2.  If  from  one-fifth  of  a  man's  money,  one-sixth  be  taken, 
he  will  have  one  dollar  left :   how  much  has  he  % 

Let   X   denote  the  amount  which  he  has  : 
Then  by  the  conditions  of  the  question 

XX 

Multiplying  both  members  of  the  equation  by  30,  the 
least  common  multiple  of  the  denominators,  and  we  have 

Qx  —  bx  =  30,    whence,     x  =  30. 

3.  John  sells  one-third  of  his  eggs,  and  then  one-half  of 
what  he  first  had,  after  which  he  has  three  left :  how  many 
had  he  at  first? 

4.  John  gave  one-third  of  his  apples  to  Charles,  and 
Charles  gave  one-fourth  of  what  he  received  to  William, 
and  then  had  six  left:  how  many  apples  had  John? 

5.  If  a  certain  number  be  diminished  by  three,  and  one* 
third  of  the  remainder  be  subtracted  from  the  number,  the 
result  will  be  equal  to  eleven :  what  is  the  number  1 

6.  The  diflference  between  five-sixths  of  a  number  and  one- 
tljird  of  the  same  number  is  nine     what  is  the  number  ? 


IN  TROD  UCTIO  K.  24) 

7.  The  diffcronce  between  four-fifths  of  a  number  and  on©, 
third  of  a  nunil)er  is  seven  :  what  is  the  number? 

8.  If  from  five-eighths  of  a  number  we  take  one  half  the 
numl)er,  and  then  take  one  from  the  difTerence,  the  result  wilJ 
be  equal  to  nothing  :  what  is  the  number  ] 

9.  A  market  woman  bought  a  certain  number  of  eggs, 
ne-lhird  of  which  she  sold  :  five  of  the  eggs  spoiled,  and 

♦he  then  had  just  three-quarters  of  a  dozen  left  :  how  many 
did  she  buy? 

10.  The  ditference  between  one-half  of  a  number  and  one- 
<ifth  of  it  is  three  :  what  is  the  number  ? 

IJ.  What  is  the  value  of  2:,  in  the  equation 
^        ^        ^       ^        ^ 

12.  What  is  the  value  of  x,  in  the  equation 

X  X  1 


18.  What  is  the  value  of  ;r,  in  the  equation 


X  X 


LESSON    XVI. 

a  laborer  can  do  a  piece  of  work  in  five  days,  what 
ft  of  it  can  he  do  in  one  day  ? 

2.  If  James  can  do  a  piece  of  work  in  eight  days,  how 
much  of  it  can  he  do  in  one  day  ?  How  much  in  two  daysl 
How  much  in  three  days  ?     How  much  in  x  days  "? 

3.  James  can  do  a  piece  of  work  in  three  days,  and  John 
do  it  in  six  days:  in  how  many  days  can  they  both  do 

It,  workirg  togother  ? 


K 


30  ELEMENT ARTALGEBRA. 

Let        1   denote  the  work  to  be  done  ; 
and  X  =  the  time  in  which  both  can  do  It,  together 

Then,       —  =   what  James  can  io  in  one  day, 
•        o 

X 

and  —-  =  what  James  can  do  in  x  days  •. 

alio,        -r  =  what  John  can  do  in  one  day, 
and  —  =  what  John  can  do  in  x  days. 

X  X 

Then^      —  f  —  =  1,  the  work  done; 
o  u 

and  by  clearing  the  equation  of  fractions, 

2^:  -h  a;  =  6,    or     3x  =  6 ; 

hence,        a?  =  —  =  2. 

o 

Therefore,  together,  they  can  do  the  work  in  two  days. 

4.  If  A  can  do  a  piece  of  work  in  f(jur  days,  and  R  can 
do  the  same  work  in  twelve  days,  how  long  will  it  take 
both  of  them  to  do  the  same  work  ? 

Let  X  denote  the  time  :  then  by  the  conditions  of  the 
question, 

whence,  x  is  found  equal  to  3. 

5.  If  Charles  can  do  a  piece  of  work  in  five  days,  and 
John  ill  twenty  days  :  how  long  will  it  take  both  of  them, 
working  together,  to  do  the  same  worki 

6.  A  barrel  can  be  emptied  by  one  faucet  in  six  hours, 
ind  by  another  in  thirty  hours  :  how  long  will  it  Lake  both 
to  empty  it,  running  together  ? 


INTRODUCTION. 


31 


Then, 

t 


7.  A  hogshead  can  be  emptied  by  one  faucet  in  seven 
rs,  and  by  another  in  forty -two  hours  :  how  long  will  it 
e  both  to  empty  it,  running  together  ] 

8.  If  A  can  do  a  piece  of  work  in  four  days,  B  in  fi\re 
ys,  and  C  in  six  days ;  in  how  many  days  will  ihey  per- 
m  it,  when  working  together  ? 

Let       1   denote  the  work  to  be  done  ; 
X  denote  the  number  of  days. 


and 


Also, 


and 


—  =   what  A  can  do  in  one  day ; 

z 
— -  =   what  he  can  do  in  x  days ; 

5 

X 

y 

-;-  =   what  C  can  do  in  one  day, 


=   what  E  can  do  in  one  day, 
=  what  he  can  do  m  x  days. 


—  =  what  C  can  do  in  a*  days. 


Then,  by  the  conditions  of  the  question, 

X  X  X  ^ 

\he  multiplying  by  60,  the  least  common  multiple  of  the 
denomiriiitors,  we  have 

15a; -I-  l2x+  10j:  =  60: 


V  hence. 


9.  An  orchard  of  pear  and  cherry  trees  has  twenty  tree* 
of  both  sorU  :  if  the  number  of  pear  trees  be  diminished 
by  twice  the  number  of  cherry  trees,  the  remainder  will  bf 
equal  to  5  :  how  many  are  there  of  each  sort  1 


32  ELEMENTARY     ALGEBRA. 

10.  James  bought  a  pencil  and  a  knife,  for  which  he  paid 
one  dollar :  what  he  paid  for  the  pencil,  diminished  by  twice 
what  he  paid  for  the  knife,  is  equal  to  minus  twenty  :  what 
did  he  pay  for  each  1 

11.  Divide  twenty-four  into  two  such  parts  that  the 
greater  diminished  by  twice  the  less  shall  be  equal  to  :he 
less  :  what  are  the  parts  1 

12.  James  and  John  together  have  twenty  oranges.  Fou/ 
times  John's  oranges  taken  from  twice  James',  leaves  a 
remainder  equal  to  half  the  whole  number  of  oranges:  how 
many  had  each  1 

13.  James  buys  a  number  of  oranges.  lie  gives  three 
away,  and  then  divides  the  remainder  equally  amT)ng  eight 
boys.  Now,  the  whole  number  of  oranges  diminished  by 
the  share  of  each  boy,  is  equal  to  seventeen  :  how  many 
oranges  did  he  buy  1 

14.  The  difference  between  a  father's  age  and  his  son' 
age  is  24  years.     But  if  the  father's  age  be  diminished  by 
twice  the  son's  age,  the  remainder  will  be  four :  what  is  the 
age  of  the  father  1 

15.  A  drover  scld  one-third  of  his  cattle  to  one  man,  and 
one-third  of  the  remainder  to  another,  and  then  had  sixteen 
left :  how  many  had  he  at  first  1 

16.  A  man  goes  to  a  tavern,  where  he  spends  three  shil- 
lings:  he  then  borrows  as  much  as  he  has  left,  and  finds 
that  the  amount  in  his  purse  is  more  than  what  h*^  had  fit  fir^% 
by  four  shillings  :  how  much  had  he  at  first  ] 


EIEMENTARY    ALGEBRA. 


CHAPTER  1. 

Preliminary  Definitions  and  Remarks, 

Quantity   is   a  general  term   applied  to  every  thing 
^hich  can  be  increased  or  diminished,  estimated  or  measured. 

2.  Mathematics  is  the  science  of  quantity. 

3.  Aloebra  is  that  branch  of  mathematics  in  which  the 
^^Mntitics  considered  are  represented  by  letters,  and  the 
^^^erations  to  be   performed   upon   them  are  indicated  by 

signs.     These  letters  and  signs  are  called  symbols. 

4.  The  sign  +,  is  called  plus  ;  and  indicates  the  addition 
of  two  or  more  quantities.  Thus,  9  +  5,  is  rf'ad,  9  plus  5, 
or  9  augmented  by  5. 

If  we  represent  the  number  nine,  by  the  letter  a,  and 
the  number  5  by  the  letter  6,  we  shall  have  a  +  i,  which  is 
read,  a  plus  b  ;  and  denotes  that  the  number  represented  by 
d  is  to  be  added  to  the  number  represented  by  b. 

5.  The  sign  — ,  is  called  minus ;   and  indicates  that  oiie 


1.  What  is  quantity  t 

2.  What  ia  Mathematics! 

8  "VN'lmt  is  Algebra  ?     "Wliat  are  the  letters  and  signs  called  I 

4.  What  does  the  sign  plus  indicate? 

5.  What  does  the  sign  minus  indicate  t 


34  ELEMENTARY     ALGEBRA. 

quantity  is  to  be  subtracted  from  another.     Thus,  9  —  5  la 
read,  9  minus  5,  or  9  diminished  by  5. 

In  like  maimer,  a  —  6,  is  read,  a  minus  i,  or  a  dii  Jnished 
by  6. 

6.  The  sign  X ,  is  called  the  sign  of  multiplication  ;  ai»d 
jrhftn  placed  between  two  quantities,  it  denotes  that  they 
»rfc  to  be  multiplied  together.  The  multiplication  of  two 
quantities  is  also  frequently  indicated  by  simply  placing  a 
point  between  them.  Thus,  36  x  '^5,  or  3(i2a,  is  read,  3«i 
multiplied  by  25,  or  the  product  of  36  by  25. 

7.  The  multiplication  of  quantities,  which  are  represeuccd 
by  letters,  is  indicated  by  simply  writing  the  letters  one  atter 
the  other,  with(»ut  interposing  any  sign. 

Thus  ab  signifies  the  same  thing  as  a  x  b-,  or  as  a.h  ; 
and  abc  the  same  as  a  X  ^  X  c,  or  as  a.b.c.  Thus,  if  we 
suppose  a  =  30,  and  b  =  25,  we  have 

a6  =  36  X  25  =  900. 

Again,  if  we  suppose  a  =  2,  6  =  3  and  c  =;  4,  we  have 

a6c  =  2  X  3  X  4  =  24. 

It  is  most  convenient  to  arrange  the  letters  of  a  product 
in  alphabetical  order. 

8.  In  a  product  denoted  by  several  letters,  as  abc,  the 
single  letters,  a,  6,  and  c,  are  called  literal  factors  of  the 
product.  Thus,  in  the  product  «/>,  there  are  two  literal  fac- 
tors, a  and  b  ;  in  the  product  abc,  there  are  three,  a,  6,  and  c. 

6,  What  is  the  sign  of  multiplication  ?  What  does  the  sign  of  multi 
pjication  indicate  ?     In  how  many  ways  may  multijiHcation  be  expressed 

7,  If  letters  only  are  used,  how  may  their  multiplicatioi  be  expre.«.««'d 

8,  In  the  {troduct  of  several  letters,  what  is  each  letter  called  ?  How 
•»muy  fkctoi's  in  obi — lu  abc? — lu  abed? — In  abcdf? 


y 


DBFINITIOS     OF     TERMS.  35 


H9.  There  are  three  signs  used  to  denote  division.     Thus, 
I       a  -^  b  denotes  that  a  is  to  be  divided  by  6. 


a 

-  denotes  that  a  is  to  be  divided  by  b. 


a  I  b      denotes  that  a  is  to  be  divided  by  b. 

10.  The  sign  =,  is  called  the  sign  of  equality,  and   ia 

»,  is  equal  to.     When  placed  between  two  quantities,  it 
►tes  that  they  are  equal  to  each  other.    Thus,  9  —  5  =  4 : 
is,  9  minus  5  is  equal  to  4 :  Also,  a  +  b  =  c,  denotes 
the  sum  of  the  quantities  a  and  h  is  equal  to  c. 
we  suppose  a  =  10,  and  6  =  5,  we  have 
a+-b  =  c,     and     10 -f  5  =  c  =  15. 

11.  The  sign  >,  is  called  the  sign  of  ineqnalifi/,  and  is 

Iu^rd  to  expreiis  that  one  quantity  is  greater  or  less  than 
Bther. 

Thus,  a  >  6   is  read,  a  greater   than   b  ;    and   c  <  rf  is 
;<1,  c  less  than  rf;    that  is,   the   opening  of  the  sign   is 
turned  towards  the  greater  quantity.     Thus,  if  a  =  9,  and 
6  =  4,  we  write,  9  >  4. 

12.  If  a  quantity  is  added  to  itself  several  times,  as 
a4"a-|-«4-«4-o  +  «,  we  generally  write  it  but  once,  and 
tlK'U  place  a  number  before  it  to  show  huw  many  times  it 
i:j  taken.     Thus, 

a  +  a  -\-  a  -\-  a  -\-  a  :=  ba. 

How  many  signs  are  used  in  division !     Wliat  are  they  I 
.   What   ia  tlie   sign   of   equality  I      When    placed    between    two 
ititieB,  what  does  it  indicate  ? 

For  what  is  the  sign  of  inequality  used!  Which  quantity  is 
(1  on  the  side  of  the  opening  ? 

What  ia  a  co-tfficient  ?  How  many  times  is  ab  taken  in  the  ex- 
on  ab\  In  Zabi  In  4<i6?  In  5a6?  In  6a6f  If  no  coefliaeut 
itlen,  what  co-efficic^t  is  luiderstiod  ? 


86  ELEMENTARYALOEBRA. 

The  number  5  is  called  the  co-efficient  of  a,  and  denotes 
that  a  is  taken  5  times. 

U  the  CO- efficient  is  1,  it  is  generally  omitted.  Thus^  a 
and  1  a  are  the  same,  each  being  equal  to  a,  or  to  one  a. 

13.  If  a  quantity  be  multiplied  continually  by  itself,  as 
aX«XaXaX«,  we  generally  express  the  product  by 
v,riting  the  letter  once,  and  placing  a  number  to  the  light 
of,  and  a  little  above  it :  thus, 

aXaX«XaX«=a*. 

The  number  5  is  called  the  exponent  of  a,  and  denotes 
the  number  of  times  which  a  enters  into  the  product,  as  a 
factor.  For  example,  if  we  have  a?^  and  suppose  a  =  3, 
we  write, 

a?  =  aX  a    Xa   =33  =  3x3x3  =  27. 
If  a  =  4,  a3  =  43  =  4   X  4  X  4  =  64, 

aud  for  a  =  5,       a^  =  5^  =  5    x  5  X  5  =  125, 

If  the  exponent  is  1,  it  is  generally  omitted.  Thus,  a^  is 
the  same  as  a,  each  expressing  that  a  enters  but  once  as  a 
factor. 

14.  The  poiver  of  a  quantity  is  the  product  which  results 
from  multiplying  that  quantity  by  itself  a  certain  number 
of  times.     Thus, 

a3  =  43  =  4  X  4  X  4  =  64, 
64  is  the  third  power  of  4,  and  the  exponent  3  shows  the 
deg'^t'  of  the  pov/er. 

15.  The  sign  -y/     ,  is  called  the  radical  sign,  and  when 

13i  What  does  the  exponent  of  a  letter  denote  ?  How  many  times 
is  a  factor  in  a^  ?  In  aM  In  a*  ?  In  aM  If  no  exponent  is  written, 
what  exponent  is  understood  ? 

14.  What  is  the  power  of  a  quantity?  What  is  the  third  powei 
of  2  ?     Express  the  fouith  power  of  a \ 

15t  Express  the  square  root  of  a  quantity?  Also  the  cube  root 
Alao  the  4th  root 


DEFINITION     OF     T! 


MS. 


tn 


Led. 


to  a  quantity,  indicates  that  its  root  is  U 
Thus, 


e  eX' 


^/a~or  simply  -y/a~denotes  the  square  root  of  a, 
^/a~ denotes  the  cube  root  of  a. 
^/a  denotes  the  fourth  root  of  a. 
The  number  placed  over  the  radical  sign,  is  called  the  iV 

»of  the  root.     Thus,  2  is  the  index  of  the  square  root,  3 
he  cube  root,  4  of  the  fourth  root,  d:c. 
'  we  suppose  a  =  04,  we  have 

-v/C4  =  8,  ^04  =  4. 

16.  Every  quantity  written  in  algebraic  language,  that 
is,  with  the  aid  of  letters  and  signs,  is  called  an  algebraic 
quantity^  or  the  aleyebraic  expression  of  a  quantity.    Thus, 

is    the    algebraic    expression   of   three 

times  the  number  a ; 
is  the  algebraic  expression  of  five  times 

the  square  of  a  ; 
is  the    algebraic    expression  of   seven 

times  the  product  of  the  cube  of  a  by 

the  square  of  b ; 
is  the  algebraic  expression  of  the  differ- 

encc  between  three  times  a  and  five 

times  h ; 
is  the  algebraic  expression  of  twice  the 

square  of  a,  diminished  by  three  times 

the  product  of  a  by  6,  augmented  by 

four  times  the  square  of  b. 
1     Write  three  times  the  square  of  a  multiplied  by  the 
cube  of  b.  Ans.  Sa'^b^ 

\ii.  What  is  an  algebraic  quantity  ?  Is  6ab  an  algebraic  quoiitityl 
Is  va       A3  1 1/1     U '6b  —  xi     Give  oUier  examples. 


2a'  -  Sab  +  4b^ 


38  ELEMENTARY     ALGEBRA. 

2.  Write  nine  times  the  cube  of  a  multiplied  by  6,  dimin 
ished  by  the  square  of  c  multiplied  by  d.       An.s.  i)a^O  —  cM 

3.  If  a  =  2,  6  =  3,  and  c  —  5,  what  will  be  the  value  of 
3a2  multiplied  by  6^,  diminished  by  a  multiplied  by  b  muJ 
tiplied  by  c.     We  have 

3a262  -  a6c  =  3  X  22  X  32  -  2  X  3  X  5  =  78. 

4.  If  a  =  4,  5=6,  c  =  7,  d  =  S,  what  is  the  value  of 
Oa^  +  hc-  ad'i  An,.  154. 

5.  if  a  =  7,  6  2=  3,  c  =  7,  G?  =  1,  what  is  the  value  of 
Qad  -f-  362c  -  4c/*''  1  Ans.  227 

6.  If  a  =  5,  6  =  6,  c  =  6,  (/  =:  5,  what  is  the  value  of 
9abc  —  Sad  -f-  46c  1  ^/^5.  1564. 

7.  Write  ten  times  the  square  of  a  into  the  cube  of  b  into 
c  square  into  the  cube  of  d. 

17.  When  an  algebraic  quantity  is  not  connected  with 
any  other,  by  the  sign  of  addition  or  subtraction,  it  is  called 
a  r/ionor/dal,  or  a  quantity  composed  ol  a  single  term,  or  sim- 
ply, a  terin.     Thus, 

3a,     5a2,     >7aW, 
are  monomials,  or  single  terms. 

18.  An  algebraic  expression  composed  of  two  or  more 
parts,  connected  by  the  sign  +  or  — ,  is  called  Bl  polynomial^ 
or  quantity  composed  of  two  or  more  terms.    For  example, 

3tt  —  bb     and     2«2  —  3c6  -f  46^ 

are  polynomials. 

19.  A  polynomial  composed  of  two  terms,  is  called  a 
binomial ;  and  one  of  three  terms,  is  called  a  trinjmiaL 

17.  What  is  a  monnmial  ?     Is  Zab  a  monomial  ? 

18.  What  ia  a  polynomial  ?     Is  3a  —  6  a  polynomial  ? 

19.  What  ia  a  binomial  ?     What  is  a  trinomial  ? 


k- 


DKFINITIUN'OF     TISRM8.  30 


1^^ 


m- 


20.  Each  of  the  literal  factors  which  compose  a  term  it 
lied  a  diiiiensioii  of  the  term  :  and  the  dejiet  of  a  term  ia 
,e  number  of  these  factors  or  dimensions.     Thus, 

j  is  a   term  of  one  dimension,  or   of  the 

(      first  degree. 

j  is  a  term  of  two  dimensions,  or  of  the 

(      second  degree. 

^  --  _     _,       ,      i  is  of  six  dimensions,  or  of  the  sixth  de* 
la^h(^^laaahcc  { 

I      gree. 

21.  A  polynomial  is  said  to  be  homogeneous^  when  all  ita 
terms  are  of  the  sanje  degree.     Thus,  the  polynomial 

I         3a  —  26  -h  c    is  of  the  first  degree,  and  homogeneous. 
—  4a6  -f  6^    is  <jf  the  second  degree,  and  hc»mogeneous. 
V  —  Ac^  -\-  2c'^d    is  of  the  third  degree,  and  humogcneous. 
Sa^  -h  4aA  -\-  c    is  not  homogeneous. 

22.  A  vinculum,  or  bar  ,  or  a  parenthesis  (  ), 

is  used  to  express  that  all  the  terms  of  a  polynomial  are  to 

considered  together.     Thus, 

a  -h  6  -f-  c  X  6,    or    (a  -\-  b  -\-  c)  X  b, 
denotes,  that  the  trinomial  a  4-  6  +  c,  is  to  be  multiplied  by  b\ 
also,  a  +  bT~c  X  c-\-d-\-/,  or  {a -\-  b  ■\- c)  x  {c  -[-(/+/), 
denotes  that  the  trinomial  a  -f  ^  +  <^5  's  to  be  multiplied  by 
the  trinomial  c  -{■  d  -^f. 

When  the  parenthesis  is  used,  the  sign  of  multiplication 
is  usually  omitted.     Thus, 

(a  -f-  6  4-  r)  X  6    is  the  same  as    (a  -j-  b  -\-  c)b. 

20i  What  is  the  diinensi<»n  of  a  term  t  What  is  the  degree  of  a 
kenn  ?  How  man}  factors  in  3a6c  t  Which  are  they  K  What  is  ita 
degree  ? 

21.  When  is  a  polynomial  homngeneoutj  ?  Is  the  polynomial 
lt%'6  +  3«x'^»  homogeneous  f     Is  2a*b  —  6'  f 

22<  For  wnat  is  the  vinculum  or  bar  used  ?  Can  you  express  iht 
tsaue  with  the  pareniheaia  t 


40  ELEMENTARx      ALGEBRA. 

23.  If  two  or  more  terms  of  a  polynomial  contain  the 
same  letters,  and  the  same  letter  in  each  have  the  same  ex 
ponent.  such  are  called  similar  terms. 

Thus,  in  the  polynomial 

lab  +  ^ab  -  ^aW  +  ^a%\ 
the  terms  7a6,  and  3a6',  are  similar:  and  so  also  are  the 
terms  —  Aa'^b'^  and  Sa^fi^,  the  letters  and  exponents  in  both 
being  the  same.  But  in  the  binomial  8a^6  +  lab"^^  the 
terms  are  not  similar ;  for,  although  they  are  composed  ot 
the  same  letters,  yet  the  same  letter  in  each  is  not  affected 
with  the  same  exponent. 

REDUCTION    OF   ALGEBRAIC    EXPRESSIONS. 

24.  The  simplest  form  of  a  polynomial,  is  an  equivalent 
expression  containing  the  fewest  terms  to  which  it  can  be 
reduced.  When  a  polynomial  contains  similar  terms,  it 
may  be  reduced  to  a  simpler  form. 

1.  Thus,  the  expression  Zab  -j-  2a6,  is  evidently  equa) 
to  5a  6. 

2.  Reduce  the  polynomial  Zac  -f  9ac  +  2ac  to  its  sim- 
plest form.  Ans.  14ac. 

3.  Reduce  the  polynomial  abc  -f  4a6c  +  5a6c  to  its  sim 
pi  est  form. 

In  adding  similar  terms  together  we  abc 

take  the  sum  of  the  co-efficients  and  4a6c 

annex  the  literal  part.    The  first  term,  habc 

abc,   has    a    co-efficient  1   understood,  Idabc 
(Art.  12). 

23.  "What  are  similar  terms  of  a  polynomial  ?  Are  Za^b  and  Ga"*' 
eimilar?     Are  2a^6"  and  2a^b^'i 

24.  What  is  the  simplest  form  of  a  polynomial  ?  If  the  terms  are 
pijsitive  and  simil'W,  may  they  be  reduced  to  a  simpler  form  ?  In  what 
way  ? 


D  1£  F  I  X  1  T  I  O  N      O  F     T  E  K  M  8  .  41 

25.  Of  the  different  terms  which  compose  a  polynomial, 

Kme  are  preceded  by  the  sign  -{-,  and  the  others  by  the 
^n  — .  The  former  are  called  additive  terms,  the  latter, 
Itractive  terms. 

When  the  first  term  of  a  polynomial  is  not  preceded  by 
Auy  sign,  it  is  understood  to  be  affected  with  the  sign  -f-. 

11.  John  has  20  apples  and  gives  5  to  William:    how 
any  has  he  left  % 
Now,  let  us  represent  the  number  of  apples  which  John 
IS  by  a,  and  the  number  given  away  by  b :  the  number  he 
IS  left  will  then  be  represented  by  a  —  6.  ^ 

2.  A  merchant  goes  into  trade  with  a  certain  sum  of 
oney,  say  a  dollars ;  at  the  end  of  a  certain  time  he  hav 
lined  b  dollars :  how  much  will  he  then  have  1 
If  instead  of  gaining,  he  had  lost  6  dollars,  how  much 
would  he  have  had]  Ans.  a—  b  dollars. 

^^    Now,  if  the  losses  exceed  the  amount  with  which  he 
^Began  business,  that  is,  if  b  were  greater  than  a,  we  must 
prefix  the  minus  sign  to  the  remainder  to  show  that  the 
quantity  to  be  subtracted  was  the  greatest. 

Thus,  if  he  commenced  business  with  $2000,  and  lost 
$3000,  the  true  difference  would  be  —  $1000 :  that  is,  the 
subtractive  quantity  exceeds  the  additive  by  $1000. 
^H  3.  Let  a  merchant  call  the  debts  due  him  additive,  and 
^^■le  debts  he  owes,  subtractive.  Now,  if  he  has  due  him 
^HgOO  from  one  man,  $800  dollars  from  another,  $300  fj  om 
^Hnother,  and  owes  $500  to  one,  $200  to  a  second,  and  $50 
^^o  a  third,  how  will  the  account  stand  1      Ans.  $950  due  him. 

26«  What  are  the  terms  called  which  are  preceded  by  the  sign  -h  I 
KhiX.  are  the  terms  called  which  are  preceded  by  the  sign  —  f  If  no 
lign  is  prefixed  to  a  terra,  what  sign  is  understood  ?  If  some  of  the 
terms  are  additive  and  some  subtractive,  may  they  be  reduced  if  simi- 

rthe  rule  for  reducing  them.     Docs  the  reduction  affect  tbti 


a 


42  ELIiMENTAUy      ALOKliKA.. 

4.  Reduce  to  its  simplest  form  the  expression 

8a26  4-  5a26  -  2,aV>  -f  Aa'-b  -  ija'b  -  a^b. 
Additive  terms.  Sublractive  terms, 

-f    3a26  -    Za^b 

+    ba?b  -    Qa^b 

+    4.a^b  -      a'b 

Sum  +  12a^6  Sum  ~  lOa^ft. 

But,  12a26  ~  10a26  =  2a2^^. 

Hence,  for  the  reduction  of  the  similar  terms  of  apoljno 
mial  we  have  the  following 

RULE. 

I.  Add  together  the  co-efficients  of  all  the  additive  terras^ 
and  annex  to  their  sum  the  literal  part ;  and  form  a  single 
svhtractive  term  in  a  similar  manner. 

II.  Then^  sid)tract  the  less  co-efic lent  from  the  greater^ 
and  to  the  remainder  prefix  the  sign  of  the  greater  co- 
efficient, to  which  annex  the  literal  part. 

Remark. — It  should  be  observed  that  the  reduction.  afTects 
only  CO- efficients,  and  not  the  exponents. 

EXAMPLES. 

1.  Reduce  to  its  simplest  form  the  polynomial 
-f  2a^c^  -  Aa^c^  -f  Ga^bc^  -  Sa^bc^  +  lla  U\ 
Find  the  sum  of  the  additive  ar  d  subtractive  terms  septv 
rately,  and  take  their  difference :  thus, 

Additive  terms,  Subtractive  terms, 

-f    2a^c^  —   4a^bc^ 

+    Qa^c'^  -  Sa^bc^ 

+  lla^^c'^  Sum    -"r2a-ig^ 

Sum  +  Ida^c^ 

Hence,  we  have,  lOa^bc^  —  I2a^bc'^  ~  7a^bc\ 


43 


2.  Reduce  the  polynomial    40^6  —  Sa^ft  —  Oa^J -|-  lla^ft 
its  aimplest  form.  Ans.   —  2a^b. 

3.  Reduce  the  polynomial    labc^  —  abc^  —  labc^  -f  Sabc^ 
(jabc^   to  its  simplest  form.  Ans.       ISaftc^. 

4.  Reduce   the    polynomial     9cP  —  8ac^  -|-  Ibcb^  +  Sea 
9ac^  —  2-ki»3   to  its  simplest  form.  Ans.  ac^  -\-  Sea, 

llie  reduction  of  similar  terms  is  an  operation  peculiar  to 
jcbra.     Such  reductions  are  constantly  made  in  Algebraic 
Addition,  Subtraction,  Multiplication,  and  Division. 


ADDITION. 

Idition  in  Algebra,  is  the  process  of  finding  thf 
iinplest  equivalent  expression  fur  several  algebraic  quan 
ties.     Such  equivalent  expression  is  called  their  sum. 

1.  What  is  the  sum  of 

Zax  +  ^ab   and    +2az  +  ab. 

3aar  +  2a6 
We  reduce  the  terms  as  in  Art.  25,  —  2ax  -j-  ab 
id  find  for  the  sum  ojr  -f-  3o6 


2.  Let  it  be  required  to  add  together 
16  expressions : 


3a 
56 
2c 


The  result  is 3a  -f  56  -f  2c 

expression  which  cannot  be  reduced  to  a  more  simpl«s 


36i  Wliat  is  addition  in  Algebra  f      What  \n  sucL  simplest  aud  rqui 
aUnt  expreaaiuD  called  f 


44  ELEMENTARY     ALGEBRA. 

(  4a263 

Again,  add  together  the  monomials  J  2a^^ 


The  result  after  reducing  (Art.  25),  is  .     .  l^a'^b^ 

f  2a2  —  4a6 
3.  Let  it  be  required  to  find  the  sum  jq2_Qj_j_    52 
of  the  expressions  J  ^  , ,2 

Their  sum,  after  reducing  (Art.  25)  is  .  ba^  —  bah  —  W- 

27.  As  a  course  of  reasoning  similar  to  the  above  would 
apply  to  all  polynomials,  we  deduce  for  the  addition  of 
algebraic  quantities  the  following  general 

RULE. 

I.  Write  down  the  quantities  to  he  added  so  that  the  similar 
tirms  shall  fall  in  the  same  column,  and  give  to  each  term  its 
projter  sign. 

II.  Reduce  the  similar  terms,  and  after  these  results,  write^ 
with  their  'projper  signs,  the  terms  which  cannot  be  reduced. 

EXAMPLES. 

1.  What  is  the  sum  of  Saa-,  5a:r,    —  2aa;,  and  13aa;.1 

Ans.   \^ax, 

2.  What  is  the  sura  of  4a6  -f  8ac  and  2ab  —  lac  -\-  dl 

Ans.  (jab  -j-  ac -\-  d 

3.  Add  together  the  polynomials, 

3a2  —  2b^  —  4a&,  5a2  —  62  -|_  2ab,  and  3a6  —  3c2  -  262. 
The  term  3a2  being  similar  to 
5a2,  we  write  8a^  for  the  result 
of    the    reduction    of    these   two^ 
terms,  at  the  same  time  slightly 
crossing  them,  as  in  the  first  term. 

SJ7.  Give  Uie  rule  for  the  addition  of  Algebraic  (juautitiea 


3^2  _  4^5  _  262 
5ii^-^2'^^b-    62 

+  3^6  -  262  _  3^2 
8a2+    a6  -  562^-"3c"2 


AUDI TION. 


45 


ig  then  to  the  term  —  4a6,  which  is  similar  to  +  2a6 
-f  3a6,  the  three  reduce  to  -f  a6,  which  is  placed  after 
and  the  terms  crossed   like  the  first  term.     Passing 
m  to  the  terms  involving    6^^      e  find  their  sum  to  be 
562,  after  which  we  write   —  Sc^. 
[The  marks  are  drawn  across  the  terms,  that  none  of  them 
ly  be  overlooked  and  omitted. 


(4)             (5) 

(^) 

(7) 

(8) 

a                 Qa 

5a 

3a6 

Sac 

a                 5a 

56 

5a6 

8a6 

8ac 

2a               Ua 

5a +  56 

Mac 

(9) 

(10) 

(11) 

7a6c  +  Qax 

8aa:  -f  36 

12a-    Oc 

-  3a6c  —  3aa; 

5ax  -  96 

- 

-3a-    9c 

4a  6c  -h  ^^ 


13ax  —  66 


9a  —  15c 


Note. — If  a  =  5,    6=4,    c  =  2,  ar  =  1 ,  what   are    the 
imerical  values  of  the  several  sums  abov^  found] 


f         (12)                     ( 

9a +/               (Sax 

—  Ga  4-  ^           —  ^aa; 

-2a-/                 ax 

13  ^ 
+  1 

)                             (lO 
8ac               3a/ -1     g    -j-  m 
9:tc                 ag  -   3a/  —  m 
lac                 ah  —     ag  -\-  Zg 

a^-g                0 

0                  ah  -{   Ag 

(15) 

Ix  -f  3a6  -h    3c 

'    -^  3ar  —  3a6  —    5c 

hx-S^ah-    9c 

9ar-9a6-    lie 

(  1^^'  ) 

8a:2+    9acir-f    13^262^2 

-  7a:2-13acarH    H-.'^V 

-  4x^  +    4acx  -  S>/)a'^V2 
-3ar2+    0       ^     7^,op^ 

(17) 
22A  _3c-7/+3^ 
■    3A-h8c-2/-97+5« 
19AH-5c-9/-G^-f  5a? 

(18) 
19a;i2  -f  3c-^6*  -  8aa^ 
-  17aA2  -  9aH*  4  ^ax' 
2ah''  -  Ca^b'  4      «•' 

40  ELEMKNTAKY     ALGEBRA. 

(19)  (20) 

7a;  —  9y  +  52  -h  3  —    ^  8a  +    6 

—  X  ~Zy            — 8—    ^  2a—    h-\-    c 

^    x-^-    y  ~Zz-\-\  -\-lg  _  3a  +    6           -^  2d 

~  2a;  -\.Qy  -\.Zz  —  \  —    g  —  66  —  3c  4-  3d 

«  +  8y  —  53  +  9  —    ^  —  5a            +  7c  -  8ci 

43^T"3y"4^"0^+~4  +  5^"  2a  —  56"+~5c~"  3d 

21.  Add  together    _  6  +  3c  -  d  -  115e  +  6/-  5*/,    36 

—  2c  —  3d  —  e  +  27/,     5c  -  8d  +  3/  —  Ig,     —  76  ~  Qc 
^-  17d  +  9e  -  5/+  11<7,     -  36  —  5d  -  2c  +  6/-  9^  +  h. 

Ans.   —  86  —  109c  -h  37/—  lOy  +  h,. 

22.  Add  together  the  polynomials,    7a26  —  3a6c  —  862c 

—  9c3  +  cd?,     8abc  —  5a^b  +  Sc^  —  4b^c  -f  cd2     and     4a26 

—  8c3  +  962c  -  Sd\ 

Ans,   Ga^b  +  5a6c  —  362c  —  Uc^  +  2cd2  —  Sd\ 

23.  What  is  the  sum  of,    5a26c  -f  66a;  —  4a/,     —  3a26c 

—  66a;  +  14a/,    —af+  96a;  -f  2a26c,    +  Gaf  —  Sbx  -h  6a26c. 

Ans.   lOa'^bc  +  6a;  -|-  15a/. 

24.  What  IS  the  sum    of,    a2w2  ^  Sahn  -f  6,     —  6a2w2 

—  Oa^m  —  6,    4-  96  —  9a^m  —  5a2w2. 

Ans.   —  10a2?i2  _  I2ahn  +  96. 

25.  What     is    the    sum    of,     4a362c  —  16a*a:  —  9aa;-^ 
•f  6a^62c  —  Gax^d  -f-  17a*a;,    +  16a:<;^d  —  a'x  —  9aWc. 

Ans.  a^b^c  -|-  aar'^d. 

26.  W^hat  is  the  sum  of,    -  7^  +  36  +  4*7  -  26,   +  Sg 
-36  +  26.  An6.  0. 

27.  What   is   the   sum  of,    a6  +  3a;y  —  7n  —  n,     —  Gxy 
-3m+ll7i4-cd,    -\- Sxy  +  Am  —  lOn -{- fg. 

Ans.  ab  -f  cd  ■\-  fg. 

28.  What  is  the  sum  of,    4.ry  -V  n  -{-  6o.r  -f-  9am,     ■—  Gxy 
I  6»  —  Gax  —  Qam,   2xy  —  In  -f-  o,'^  —  «wi.       Ans,   -\-  as. 


SIBTR AC  TJON 


47 


i 


29.  Add    the    polynomials      10aV6  —  V2a^cb^     5a V6 
Uo'cO  -  lOaar,      -  2a^2-'6  -  12aV6,     and    -  ISa^x'^b 


12«\-6  -f-  9ajr. 


^1/^v.  4aV6  -  22ttV6  -  a^. 


30.  Add  together    3a  -f  6  +  c,      5a  -f  26  -f  3ac,     a  -f  c 
4-  ac^  and   —  3a  —  9ac  —  86.       Ans.  6a  —  56  +  2c  —  5a<;. 

131.  Add    together    5a-6  +  Gcx  +  96^^,     7car  —  8a26,  and 
15cx  —  96c'^  +  2a26.  Ans.   —  a^b  —  2cx. 

32.  Add  together    8ai;  +  5a6  +  3a26?c2,     -  18aa;  -f  6a« 
10a6,    and    lOaa;  —  15a6  ~  Ga'^6V. 
Ans.   —  3a262c2 -f  0a2. 
33.  Add   .together     3a2  -|-  5a26V  -  9a3jr,    Ta^  -  80^6^ 
-  lOa^a:,    and    10a6  -f-  ICa-^iV  -|-  19«''ir. 

Ans.  10a2  +  13a26V  +  10a6. 


SUBTRACTION. 

htraction,  in  Algebra,  is  the  process  of  finding  the 
►lest  expression  for  the  diflerence  between   two   alge. 
kraic  quantities. 
Thus,  the  difference  between  Ga  and  3a  is  expressed  by 

6a  -  3a  =  3a ; 
id  the  diflerence  between  7a^6  and   Sa^b  by 

7a'^6  -  3a'^6  =  4a^. 
In  like  manner,  the  difference  between    4a   and   36,    is 
expressed  by  4a  —  36.     Hence, 

1/  the  quantities  are  positive  and  similar,  subtract  the  co- 
efficients,  and  to  their  difference  annex  the  literal  part.  If 
thetj  are  not  similar,  place  the  minus  sign  before  the  quantity 
to  be  subtracted. 

28.  Wlmt  is  subtraction  j  Algebra?  How  do  you  fiml  this  differ- 
uncc  when  tlie  quantities  are  positive  and  similar  f  When  the}  are  no< 
tiiiiilar,  huvi  U )  yuu  ezprcus  tLo  differeucu  f 


48  ELEMENTARY     ALGEBRA 

(1)  (2)  (3) 

From  Sab  Qux  Qabc 

take  2ab  Sax  labc 

Rem.  ab  Sax  2abc. 

(4)  (5)  (6) 

From  IQa^^c  lla^b^c  2^aWx 

take  9a262c  Sd^b\  la^b'-x 

Rem.  To^  l4^3&3^  iW6V. 

(7)                       (8)  (9) 

From                     Sax                        ^abx  2am 

take                      8c                          9ac  ax 

Rem.         Sax  —  Sc  4:abx  —  9ac  2am  —  ax, 

29.  Let  it  be  required  to  subtract  from    4a 

the  binomial 2b  —  Sc 

The  difference  may  be  put  under  the  form  4a  —  (26  —  3c) 
We  must  now  remark  that  it  is  the  difference  between  2h 
and  3c  which  is  to  be  taken  from  4a. 

If  then,  we  write 4a  —  26, 

we  shall  have  taken  away  too  much  by  the  units  in  3c; 
hence,  3c  must  be  added,  to  give  the  true  remainder,  which 
is 4a  —  26  -h  3c. 

To  illustrate  this  example  by  figures,  suppose  a  =  5, 
6  =  5,  and  c  =  3. 

We  shall  then  have 4a  :_  20 

and 26  -  3c  =  10  -  9    r:^    1 

which  may  be  written     4a  —  (26  —  3c)  =  20  —  1    —"19^ 

29 1  If  26  —  8c  is  to  be  taken  from  4a,  •what  is  proposed  (o  be  done  I 
If  you  subtract  26  from  4a,  have  you  taken  too  much !  How  th^c 
muHt  you  supply  the  deficiency  ? 


SUBTKAC   riON. 


40 


Here  it  is  required  to  subtract  1  fiom  2l>.     If,  then,  we 
subtract   26  =  10,  from   4a  =  20,    it  's  plain  that  we  shall 

rve  taken  too  much  by    3c  =  9,    whi»;h  must  therefore  be 
ded  to  give  the  true  remainder. 


30.  Hence,  for  the  subtraction  of  algebraic  quantities,  we 
,VG  the  following  general 


RULE. 


I.  Write  the  quantity  to  be  subtracted  und^r  ihatfrom  which 
it  is  to  be  taken  J  placing  the  similar  tertns^  if  there  are  any,  in 
the  same  column. 


II.  Change  the  signs  of  all  the  terms  of  the  subtrahend,  or 
rnceive  them  to  be  changed,  and  then  reduce  the  polynornial 
fsult  to  its  simplest  form. 


EXAMPLES. 


p 

F 

p 


(1) 

From  Qac  —  bab  -\-  c* 
ake  3ac  -f-  Sab  -f  7c 
em.  3ac  —  8a6  -f-    c^  —  7c. :-  *,  ^ 


=  5.S 


(1) 


(2) 
From      Goa;  —  a  +  36» 
ake        ^ax  —  ar  +    6^ 
em.  —  3aar  —  a  4- ar-f  2^2. 


(4) 
From      5a3-4«26-|.    3^2^ 

ake  -  2.734-30^6-    W^c 

em.       7a3_7a26  4-  Wb'^c. 


(Sac 
—  Sac 

—  5a6  4-    c^ 

-  3a6  -  7c 

^        3ac 

-8a6  4-    c2-7c. 

(jyx- 
yx- 

(3) 

3x2  _{_  56 

3     4-    a 

byx  — 

3ar2  4-3  4-56~a. 

4a6  — 
5a6- 

(5) 
crf4-3aa 
4a/  4-  3a2  -|-  56». 

-  a6  4-3crf-56^. 

30*  Give  the  rule  for  the  subtraction  of  Algebraic  qaantilicti. 
3 


50  ELEMENTARY     ALGEBRA. 

6.  From    6am  +  y   take  oain  —  x.     Ans.  ^am  -{-  x  -\-  y, 
I  7.  From    Sax   take   Sax  —  y,  Ans.    -f  y» 

8.  From   7a^b^  -  x^   take    18a^^  +  x\ 

Ans.   -  na^^-~2x^ 

9.  From    -  7/  -}- Sm  —  8x   take    —  6/  —  5m  —  2ar  -f- 
3o?  -(•  8.  ^W6-.    _/  -f  8m  —  6a:  —  St/  - .  8. 

10,  From    —  a  —  56  +  7c  —  c?   take   4&  —  c  +  2c?  +  2/5;. 
^ws.   —  a  —  96  +  8c  —  8g?  —  2A:. 
^11.  From   .  .  -  3a+  b  —  8c+  7e  -  5/+  3A  -  7a;  -  IS^/ 
take  k  +  2a  —  9c  +  Se  —  7x  +  7f—y  —  Sl  —  k. 

Ans.   -5a  +  6  +  c-e-  12/+  3A  -  12y  +  3Z. 

12.  Fi'om   a  -\-  b   take   a  —  b.  Ans.  26. 

13.  From    2a;  —  4a  —  26  -f  5    take   8  —  56  +  a  +  6.x. 

Ans.   —  4a;  —  5a  -}-  36  —  3. 

14.  From   3a +  64  c  —  c?  —  10   take   c  +  2a  —  ^. 

Ans.  a  -r  b  —  10. 

15.  From   3a  +  6  +  c  —  c?  -  10   take  6  -  19  +  3a. 

Ans.  c  —  c?  +  9. 
^  1().  From    2a6  -f-  6^  —  4c  +  6c  —  6   take  3a2  —  c  +  b\ 

Ans.  2ab  —  Sa^  —  Sc  -\-  be  —  b. 

17.  From   a^  +  Sb^c  +  a62  —  a6c  take  63  +  a62  —  a6c. 

Ans.  a3  4-  362c  —  6^. 

18.  From    12a;  +  6a  —  46  +  40  take  46  -  3a  +  4a;  -f  6d 
-  10.  Ans.  8a;  -f  9a  —  86  —  Cdf  -h  50. 

19.  From  2a;  —  3a  -f-  46  -f  6c  -  50  take  9a  +  a;  -|-  66  - 
'>c  —  40.  Ans.  jc—\2a  —  26  -\-  12c  —  10. 

20.  From   6a  —  46  -  12c  +  12a;  take  2a;  -^  8a  -f  46  -  6c. 

Ans.   14a  —  86  —  6c  -f  10a;. 

21.  From    8a6c  •-  126%  -f  6cx  —  7xy    take   7c.r  —  a;y  — 
136 'a.  Am.  Sabc  -i-  b^a  — ^  ex  —  0a:y, 


SUBTRACTION.  51 

31.  Polynomials  may  be  subjected  to  certain  transformiw 
tions,  by  the  rule  for  subtraction. 

First  example,     .     .  Ga^  —  Sab    +  26*    —  26r, 

becomes Ga'^  —{Sab   —  2b^    +  26c).     . 

Second Ta^  -  8a^  —  4bh  -\-  Gb\ 

becomes 7a^  -(Sa^  +  462c  -  U% 

or,  again, 7a^  —  Sa^fi  —{ibh  —  (jb^). 

Third 8a3  -  762-  +    c     -  rf, 

becomes 8a^  —(7b^    —    c     -f  rf). 

Fourth 9Z>3  _    a     -h  3a2    —  rf, 

becomes 96^  —  (a     —  Sa"^    -{-  d). 

82.  Remark. — From  what  has  been  shown  in  addition 
and  subtraction,  we  deduce  the  following  principles. 

1st.  In  algebra,  the  term  add  does  not  always,  as  in  arith- 
metic, convey  the  idea  of  augmentation  ;  nor  the  term  snm^ 
the  idea  of  a  number  numerically  greater  than  any  of  the 
numbers  added.  For,  if  to  a  we  add  —  6,  we  have  a  —  6, 
which  is,  properly  speaking,  a  diflerence  between  the  num- 
ber of  units  expressed  by  a,  and  the  number  of  units  ex- 
pressed by  b.  Consequently,  this  result  is  numerically  less 
than  a.  To  distinguish  this  sum  from  an  arithmetical  sum, 
it  is  called  the  algebraic  sum. 

Thus,  the  polynomial  2a2  --  SaH  -f  362c  is  an  algebraic 

81.  How  may  you  change  the  form  of  a  polynomial  I 

82.  In  alg.'bra  do  the  words  add  and  *um  convey  the  same  ideas  as 
in  arithmetic !     "What  is  the  algebraic  sum  of  9  and  —  4  ?     Of  8  and 

-  2 1  May  an  algebraic  sura  ever  be  negative  f  "What  is  the  sum  of  4 
and  —  8  f  Does  the  word  subtraction,  in  algebra,  always  convey  the 
idea  of  diminution  ?     "VSHiat  i^  the  algebraic  difference  between  8  and 

—  4  f     Between  a  aud  —  6  f 


52  ELEMENTARY     ALGEBRA. 

sum  of  the  monomials  2a^,  —  Sa^fi,  +  3^^^,  with  theii 
respective  signs;  but,  in  its  numerical  acceptaiio  n,  it  is  the 
arithmetical  difference  between  the  sum  of  the  units  con- 
tained in  the  additive  terms,  and  the  sum  of  the  units  con- 
tained in  the  subtractive  terms. 

It  follows  from  this,  that  an  algebraic  sum  may,  in  the 
numerical  applications,  be  reduced  to  a  negative  nuncber,  oi 
a  number  affected  with  the  sign  — . 

2d.  The  word  subtraction^  in  Algebra,  does  not  always 
convey  the  idea  of  diminution  ;  nor  the  term  difference^  the 
idea  of  a  number  numerically  less  than  the  minuend  :  for, 
the  numerical  difference  between  +  a  and  —  b  being  a  -f  ^, 
exceeds  a.  This  result  is  an  algebraic  difference,  and  can  be 
put  under  the  form  of 

a  —  (  —  5)  z=:  a  -{-b. 


MULTIPLICATION. 

33.  If  a  man  earns  a  dollars  in  one  day,  how  much  will 
he  earn  in  6  days  1  Here  it  is  simply  required  to  take  the 
number  a,  6  times,  which  gives  Qa  for  the  amount  earned. 

L  What  will  ten  yards  of  cloth  cost,  at  c  dollars  per  yard  1 

Ans.  10c  dollars, 

2.  What  will  d  hats  cost,  at  9  dollars  per  hat  ? 

Ans.  9c?  dollars 

3.  What  will  b  cravats  cost,  at  40  cents  each? 

Ans.  406  cents. 

4.  What  will  b  pair  of  gloves  cost,  at  a  cents  a  pair  1 

33  If  a  man  earns  a  dollars  in  1  day,  how  much  will  he  earn  in  4 
days?  In  5  days?  In  8  days?  In  12  days?  If  he  earns  c  dollars  a 
^iay,  how  much  will  he  earn  in  d  days  ?     What  is  multiplication  ? 


) 


MULTIPLICATION.  53 

it  is  plain  that  the  cost  will  be  found  hy  repeatiLg  b 
times  as  tliere  are  units  in  a  :  Hence,  the  cost  is 
cents.     Hence,  we  infer  that, 

UtUiplkut'ion^  in  Ahjebra^  is  the  process  of  taking  one 
lutitt/y  called  the  multiplicand,  as  many  times  as  there  are 
its  in  another^  called  the  multiplier. 

[34,  If  a  man's  income  is  3a  dollars  a  week,  how  much 

ill  it  be  in  46  weeks  1     Here  we  must  repeat  oa  dollars  as 

man}-  times  as  there  are  units  in  46  weeks ;  hence,  the  pro- 

ict  is  equal  to 

3a  X  46  =  12a6. 

If  we  suppose  a  =  4  and  6  =  3  the  product  will  be  equal 
to  144. 

Remark. — It  is  plain  that  the  product  12a6  will  not  be 
altered  by  changing  the  arrangement  of  the  factors ;  that 
is,  12a6  is  the  same  as  a6  X  12,  or  as  6a  x  12,  or  as 
ax  12  X  6  (See  Arithmetic,  §  26). 

35.  Let  us  now  multiply  Za-b^  by  2a^b,  which  may  le 
placed  under  the  form 

3a262  X  2a26  =  3  X  2aaaabbb  ; 
111  which  a  is  a  factor  four  times,  and  6  a  factor  three  times : 
hence  (Art.  13). 

3a262  X  2a26  =  3  X  2aaaa666  =  6a*63, 

in  which,  we  multiply  the  co-efficients  together^  and  add  the 
exponents  of  (lie  like  letters, 

84.  Will  a  product  be  altered  by  changing  the  arrangement  of  the 
fJEictora  f  Is  3a6  the  eaine  as  86a  f  la  it  the  same  as  a  x  86  ?  As 
6  X  Sat 

36.  In  multiplying  monomials,  what  operation  do  you  perform  on  the 
to-f'fficienta  ?  What  do  you  do  with  the  exj  onenta  of  the  commov 
VcttfiTS  I     Wlat  is  th«  rule  for  the  multiplication  r>f  niuiioiniala  I 


54 


ELEMENTARY     A  L  G  E  li  K  A  , 


Ilencc,  for  the  multiplication  of  monomials,  we  have  the 


follow  in  t 


RULE. 


I.  Multiply  the  co-efficients  together  for  a  new  co-efficient. 

II.  Write  after  this  co-efficient  all  the  letters  which  enter 
into  the  multiplicand  and  multiplier^  affecting  each  with  an 
tor^onent  equal  to  the  sum  of  its  exponents  in  both  factors. 


EXAMPLES. 

1.  8a26c2  X  7a6t^2  _  ^QaWcH\ 

2.  2\aWcd  X  8a6c3  =  lC8a*6V(/. 

3.  Aahc  X    7c//  =    2Sabcdf, 


Multiply 
by 


(4) 

3a26 
2a^ 
6a*62 


(5) 

I2a^x 


(7) 

aHy 
_2xy^ 
2a^x'^y'^ 


(8) 

3a62c3 
_9a263c 
27a36V^ 


10.  Multiply    5^362^2  ^y  6cV. 

11.  Multiply  10a*6V  by  7ac(/. 

12.  Multiply    Qa'^bxy  by  da^bxy. 

13.  Multiply  36a867c6^5  by  20a62c3f£*. 

14.  Multiply  21axyz  by  ^aWcW^xyz, 

Ans. 

15.  Multiply  lya^i-^c  by  Qahjiy. 


(6) 

6a*yg 

ay'^z 
Qaxy'^z^, 

87aa;V 
363arty3 

2Gla6%V. 

^/i5.  30a3^»2c5^! 

^/is.  70a^b^c^d, 

A71S.  Sla^'^x'^y'^, 

Ans.  720a^bhH^ 

2AZaWcHH''y''z\ 
Ans.  lO^a^b'^cxy, 


I 


I 


MULTIPLICATION  55 

la   Multiply  •20a5^»W  by  VZa-x^i/.         Ans.  2i0a'b^cdx^i/, 

17.  Multiply  i4a*b''d'i/  by  ^OaV^x^y. 

Ans.  280a"i«cV*xV. 

18.  Multiply  8a36V  by  7-.«6xy*.  ^/i5.  5(>a'6*iry^ 

19.  Multiply  Ibaxi/z  by  ba^bcdx^y'^.     Ans.  ii"iba^Ocdx^j/h, 

20.  Multiply  Sla-^y'V^  by  {)a^bc^x^i/.       Ans.  45<Ja^6cVy'. 

21.  Multiply  2u'b'Y  h  1«"^-p-  -^''«-  JiOa<6*xy». 

22.  Multiply  G4a^//t*j:^yz  by  Sab^c\ 

Ans.  512a^6Vwi^j*yi. 

23.  Multiply  9aH^c^d^  by  i2a36V.  ^/<«.,  10«a^6-c«(^3. 

24.  Multiply  2l6aZ»Vt/»  by  Oa-^6V.         Ans.iJ-iSa'b'^c^d^, 

25.  Multiply  70a»6"c^(/ya;  by  :2a'b'c^dz^i/\ 

Ans.  840ai*6'V</3yj:3y3^ 

36.  We  will  now  consi'^.i-  the  mosi.  general  case  oi  two 
lynomials. 

Let  a  represent  the  sum  of  all  the  additive  terms  of  the 

multiplicand,  and  —  b  the  sum  of  the   subtractive   terms. 

t  c  denote  the  sum  of  the  additive  terms  of  the  multi- 

lier,  and  —  d  the  sum  of  the  subtractive  terms.     The  mul- 

plicand  may  then  be  represented  by  a  —  6,  and  the  mul- 

plier  by  c  —  d  :  It  is  required  to  take  a  —  6  as  many  timea 

Jiere  are  units  in  c  —  d. 

Let  us  first  take  a  —  6  as  many 
es  as  there  are  units  in  c.  a   —  b 

We  begin  by  writing  ac^  which  is         c    —  d 
too  great,  by  6  taken  c  times ;  for,         ac  —  be 

is  only  the  difference  between  a  —  ad  -^-bd 

«nd  b  which  is  to  be  taken  c  times.         ac  —  be  —  ad  -^  bd. 
Hence,  ac  —  be  is    the   product   of 
a  —  6  by  c. 

But  it  was  proposed  to  take  a  —  b  only  as  many  times  as 
Uicrc  are  uiiiu^  in  the  difference  bet w ecu  c  and  d :  heuce,  the 


66  ELEMENTARY     ALGEBRA. 

last  product  ac  —  be  is  too  large  by  a  —  h  taken  d  times, 
But  a  —  h  taken  d  times,  is  ad  —  hd.  Subtracting  this  pr(>- 
duct  from  ac  —  ch  (Art.  30),  and  we  have 

{a  —  h)  X  (c  —  d)  z=  ac  —  he  —  ad  -\-  hd. 

37.  Hence,  we  have  the  following  rule  for  the  signs. 

When  two  terms  of  the  multi2Jlicand  and  multijilier  an 
effected  with  like  signs,  the  corresponding  product  is  affected 
with  the  sign  -(-  ;  and  when  they  are  affected  with  contrary 
signs,  the  product  is  affected  with  the  sign  — . 

Therefore,  we  say  in  algebraic  language,  that  +  multi- 
plied by  -f,or  —  multiplied  by  — ,  gives  +  ;  —  multiplied 
by  +,  or  +  multiplied  by  — ,  gives  — . 

Hence,  for  the  multiplication  of  polynomials  we  have  the 
following 

ETJLE. 

Multiply  all  the  terms  of  the  multiplicand  by  each  term  cf 
the  multijdier,  observing  that  in  each  multiplication  like  signs 
give  plus  in  the  product,  and  unlike  signs  minus.  Then  reduce 
the  polynomial  result  to  its  simplest  form. 

EXAMPLES    IN    WHICH    ALL    THE    TERMS    ARE    PLUS. 

1.  Multiply        ....       3a2  4-     4a6  +  b^ 

by         2a   +    56 

Qa^  -f    Sa^  +  2a62 
The  product,  after  reducing,  -f  ISa^J  -f.  20ab^  +  ^h^ 

becomes      ....       Qa^  +  2S^^b^22ah^-f'b¥, 

87.  What  does  +  multiplied  by  +  give  ?  +  multiplied  by  — 
-  multiplied  ly  +  f  —  multiplied  bj  —  {  Give  the  rule  for  tbi 
iiultiplication  of  polynomiab. 


MULTIPLICATION, 


57 


3j^2.  Multiply  a;2  -h  2ax  -f  a'^  by  x -\- a. 

A71S.  ar3  -f-  3aa:2  ^  3^2^.  4.  ^s 

3.  Multiply  a-^  -r  y3  by  ar  +  y.  -i4ns.  ar*  +  xy^  -f  a;^y  +y* 

4.  Multiply  3a62  4-  6aV  by  Sab^  +  Sah\ 

H  Ajis.  da%*  +  27a362c2  +  lSa*c* 

5.  Multiply  a262  +  c'^d  by  a  +  &. 

^Be.  Multiply  3cu:2  ^  g^^^a  ^  <,^5  by  Ca^c^. 

7.  Multiply  64a3x3  +  27a2a:  +  f)ab  by  Sa^cc/. 
^B  A71S.  512a«cfl?ar2  _^  216a*crfar  -f  72a*6cd 

^'s.  Multiply  a2  +  2aa;  +  a:2  by  a  +  ic. 

^ns.  a3  -f  Sa^a;  +  Soar^  +  x^ 
9.  Multiply  a3  +  Sa^ar  +  3aa:2  4.  a:3  by  a  +  ar. 

Ans.  a*  -\-  4a^x  -\-  Ga^x^  -\-  4ax^  -\-  x* 
16.  Multiply  x^  -\-  y^  hy  x  -\-  y. 

Ajis.  x^  -\-  ary^  .4-  x^y  +  y^ 
11.  Multiply  x^  -f  xy^  +  7aar  by  ax  -\-  bax. 

Ans.  Qtai^  -f-  ^x'^y^  +  42^2x2 
12    Multiply  a3  +  Sa^fc  +  3a62  -|-  53  ^y  «  +  ft. 

^n5.  a*  4-  4a36  -f-  6a262  +  4a63  -}-  ft* 
J  3.  Multiply  a:3  +  x'^y  -f  a:y2  -f-  y3  by  a;  +  y. 

^;j5.  ar*  +  2a-3y  +  2a;2y2  +  2ary3  ^  y4 
14.  Multiply  a:3  +  2x2  -f  ar  +  3  by  3a;  H-  1. 

Ans.  3ar*  +  7a:3  ^  5^2  4.  iQa    )-  3 


GENERAL    EXAMPLES. 


V.  Multiply 
by       . 

ie  product   . 
becomes  after 

reducing     . 


a* 


2aa;  —  3aft 
3a:    ~     ft. 

6aa:2 —  9afta; 

—   2aftar  -f  .ViA» 

6aa;2— 11  aftar  4- 3a6» 


58  ELEMENTARY     ALGEBRA. 

2.  Miltiply  a*  -  2P  hj  a-b. 

Ans.  a^  —  2aP  -  a^b  +  26< 

3.  Multiply  x''-  -  3a:  -  7  by  x-  2. 

J.71S.  x^  —  5a:2  __  a;  -f-  14 
4L  Multiply  Sa^  —  6ab  -f  26^  by  a-  —  '7a,/>. 

A?is.  3a*  -  2Ga-^6  +  STa^^^  —  14(i53. 

6  Multiply  P  +  6*  +  ^'  by  62  _  i.  J^s.  b^  -  b\ 
6.  Multiply  x^  —  2^3y  ^  4;j2y2  _  8a,-y3  +  IGy*  by  a;+  2y. 

^/is.  x^  +  32?/5. 

7  Multiply  4a;2  —  2y  by  2y.  Ans.  ^x'^y  —  4^^^ 

8.  Multiply  2.r  +  4y  by  2x  —  Ay.  Ans.  4x^  —  l(jy\ 

9.  Multiply  x^  -j-  x'^y  -f-  ^y^  +  y^  by  x  —  y. 

Ans.  ar*  —  y\ 

10.  Multiply  x^  -{-  xy  -{-  y^  by  a;^  _  ^y  _|.  ^2^ 

-4// 6'.    X'*    4-  a:2i/2   _j_  y4^ 

11.  Multiply  2a2  —  3ffa;  -|-  4:X^  by  Sa^  _  Oaj;  —  2;i-2. 

Ans.  10a*  —  27a-^.i-  +  Ma'^x^  —  ISax^  —  Sx*. 

12.  Multiply  3a;2  —  2a-y  +  5  by  a-^  +  2xy  —  3. 

^A<s.  3a;*  +  4a-3y  —  4a;2  —  Ax^y"^  +  lOa-y  —  15. 

13.  Mult'ply  3a:3  _^  2x^y^  +  Sy^  by  2a;3  —  3.^2^2  +  5^3^ 

j  6a,-6  —  5xY  —  Q^Y  +  Gx'Y  + 
'  (  15xV  -  9a:2y*  _^  10^2^5  ^  15^5, 

14.  Mu^'iply  8aa;  —  Gab  —  c  by  2ax  -{-  ab  -\-  c. 

Ans.  lCa2a;2  —  4a26a:  —  Ga'^b"^  -f  Gacx  —  labc  —  c*. 

15.  MuUiply  3a2  -  5^2  +  3c2  by  a2  -  b\ 

Ans.  3a*  -  8a2Z;2  -|_  3a2c'2  +  56*  -^  •  35V. 

IG.  3a2~5M-|-    cf 

-  5a2  -j-  Ud  -  Scf. 


Pro.red.    -  15a*  -(-  S':a'bd-2Qa'^c/-20b^r--\-Ubcdf-ScY^. 


MULTIPLICATIOIC.  5V 

18.  We  will  finish  the  subject  cf  algebraic  multiplication, 

making  known  a  few  results  of  frequent  use  in  Algebra. 

Let  it  be  required  to  f^rni  the  square,  or  second  power, 

the  binomial  (a  -f-  6).     We  have,  from  known  principles, 

(a  +  by  =  (a  +  6)  (a  -f  6)  =  a2  +  2ab  f  h\     That  ia, 

e  square  of  the  sum  of  two  qvantities  is  equal  to  the  square 
the  Jirstj  plus  twice  the  product  of  the  first  by  the  second^ 
his  the  square  of  the  second. 

1.  Form  the  square  of  2a  +  Zb.    We  have  from  the  rule 
(2a   +36)2       =    4«2     _|.    i2a6     -f-    962. 

2.  (5«6  -(-  3af)2      =  25a262  +    30a26c  +    9aV. 

3.  (5a2  +  8a26)2    =  25a*     +    80a*6    +  G4a462. 

4.  {fjax  -h  9a2a;2)2  =  Z^Sa^x^  +  lOSa^x^  +  81a*i;*. 
39.  To  form  the  square  of  a  difference,    a  —  b,    we  have 

(a  -  6)'-^  =  (a  -  6)  (a  -  6)  =  a^  -  2a6  -|-  6^ :   That  is, 

The  square  of  the  difference  between  two  quantities  is  equal  to 
square  of  the  jirst^  minus  twice  the  product  of  the  first  by 
e  second,  plus  the  square  of  the  second. 

II.  Form  the  square  of  2a  —  6.     We  have 
(2a-6)2  =  4a2-4a6-|-  62. 
2.  Form  the  square  of  4ac  —  be.     We  have 
(4ac  -  6c)2  =  lGa2c2  -  8a6c2  +  62c2. 

3.  Form  the  square  of  7a^b^  —  12a63.     \Ve  have 
(7a262  -  12a6')2  ==  49a<6*  -  168a365  +  144a26«. 


A  lit 


,JtB    What  IB  the  square  of  the  sum  of  two  quantities  equal  to  f 

What  is  the  square  of  the  difference  of  two  quautities  equal  ii\\ 


00  ELEMENTARY     ALGEBRA. 

40.  Let  it  be  required  to  multiply  a  -{■  b  by  a—  h 
We  have 

{a  -\-b)  X  {a  —  b)  =  a^  —  P.     Hence, 

The  sum  of  tivo  quantities,  multiplied  by  their  difference^  u 
equal  to  the  difference  of  their  squares. 

1.  Multiply  2c  +  fi  by  2c  —  b.     We  have 

(2c  4-  6)  X  (2c  -b)  z=  4c2  -  b\ 

2.  Multiply  9ac  +  36c  by  9ac  —  36c.     We  have 

(9ac  +  36c)  (9ac  -  36c)  =  81aV  _  962c2. 

3.  Multiply  8a3  +  7a62  by  Sa^  -  7a62.     We  have 

(8a3  +  7a62)  (Sa^  _  7a62)  =  64a6  -  49a26*. 

FACTORING    POLYNOMIALS. 

41.  It  is  sometimes  convenient  to  find  the  factors  of  a 
polynomial,  or  to  resolve  a  polynomial  into  its  factors. 
Thus,  if  we  have  the  polynomial 

ac  +  a6  +  ad^ 

we  see  that  a  is  a  common  factor  to  each  of  the  terms . 
hence,  it  may  be  placed  under  the  form 

a{c-\-b  +d). 

1.  Find  the  factors  of  the  polynomial  ©252  _|_  ^^y  ^  q7J'^ 

Ans.  a\h^  -\-  ^  4-/). 

2.  Find  the  factors  of  3a26  +  6a262  +  62^. 

Ans.  6(3a2  +  Ga^o  +  bd). 


40.  What  is  the  sum  of  t\ro  quantities  multiplied  by  their  differo»«J« 
equal  to? 


DIVISION. 


01 


I 


DIVISION. 


8.  Find  the  factors  of  ^a^b  +  9a^c  +  ISa^j-y. 

Ans.  3a2(6  -h  3c  -f  ^^y), 

4.  Find  the  factors  of  Sa^cx  —  ISacx^  +  2ac*y  -  SOaSc^ar. 
Ans.  2ac{4ax  —  9z^  +  c^y  —  15aVar). 

5.  Find  the  fiKJtors  of  a^  +  2a6  +  b\ 

Am.  (a  +  &)  X  (a  +  6). 

e.  Find  the  fact<»rs  of  a^  —  b\     Ans.  {a  +  b)  x  (a  —  b). 

Kthe  factors  of  a^  —  2ab  +  b\ 
Ans.  (a  —  6)  X  (a  —  b) 
'ision,  in  Algebra,  is  the  process  of  finding,  from 
two  algebraic  expressions,  a  third,  which  being  multiplied  by 
le  second,  will  give  a  product  equal  to  the  first.     The  first 
called  the  dividend,  the  second  the  divisor^  and  the  third, 
le  quotient. 
1.  The  division  of  72a*  by  Sa^  is  indicated  thus : 

It  is  here  required  to  find  a  third  monomial,  which,  mul- 
Iplied  by  the  second,  will  produce  the  first.     It  is  plain  that 
le  third  monomial  is  da^ :  Hence 
72a5 


8a3 


9a2 ;  for,  Sa^  X  Oa^  =  72a*. 


42    What  is  diyiaion  in  Algebra?     Give  the  rule  for  dividing  mono 


The   quotient   Oa^,   is  obtained  by  dividing  the  co-efficient 
of  the  dividend  '"y  the  co-efficient  of  the  divisor,  and  subtracting 

it  exponents  of  the  common  letter. 
42    ^ 


Also, 
Again, 


ELKMENTARY     ALGEBKA 


lab 


=  ba^-'^b^-^c=5a^c, 


lab  X  ba%c  =  SbaWc, 


8a^c 


Hence,  for  the  division  of  monomials  we  have  the  fol- 
lowing 

RULE. 

I.  Divide  the  co-efficient  of  the  dividend  by  the  co-ejicieni 
of  the  divisor^  for  a  new  co-efficient. 

II.  Write  after  this  co-efficient^  all  the  letters  of  the  dividend^ 
a?id  affiect  each  with  an  exponent^  e<piul  to  the  excess  of  its 
exponent  in  the  dividend  over  that  in  the  divisor. 

From  this  rule  we  find 

=i4a^bcd'.        _  -  ba^bxd. 

VZab'^c  30a^Z;V2 


1.  Divide 

2.  Divide 

3.  Divide 

4.  Divide 

5.  Divide 

6.  Divide 

7.  Divide 

8.  Divide 

9.  Divide 

10.  Divide 

11.  Divide 


10x2  by    8X. 

Ibax^y"^  by  Zay, 
MaU^x  by    1262. 
3Ga4|(/V  by  \}a-^b'^c. 
8  a^b\  by   8a26. 
OQa^i^x*  by    lU^b'^x^, 
108x6^523  by    54^5^. 

64a:''y^2;*^  by   ICx^yV. 
96a'66c5  by  \2a%c, 
54a'c^£/^  by  21acd. 
38tt*66fi*by  2a36*J. 


Am.  2x. 
Ans.  5r2y2 
Ans.  lubx 
A /IS.  4ab-^c. 
A  ns.  1 1  aba, 
Ans.  dab^x. 

A/IS.  2xyV. 
Ans.  Axyz, 

Ans.  Sa^b^c^. 

Ans.  2aVJ5. 

Ans.  \9alHl3 


U  I  V  I  8  I  O  N , 


03 


A  lis.  (\abc. 

Ans.  2a A V. 

Ans.  Ha*x^i/\ 

Ans.  (jiJbd/^ 

Ans.  IGab^c^iT. 

Atis.  4amn. 

Ans.  hx^yH. 

Ans.  Sa*bc. 

Ans.  2a^j/z. 

Ans.  8a'^6V. 


12  Divide  42a^bh^  by  labc. 

13.  Divide  G4a-'//c8  by  S'M^bc. 

14.  Divide   I'ZSa^x^y''  by  lijaxt/*. 
Divide  1326(/y6  by  2(1*/. 

6.  Divide  25Ga^6»c8(r  by  IGa^c^. 

17.  Divide  200(i^;/i^M2  by  50a"7W7i. 

■hB.  Divide  300xVz2  by  GOxy^z, 

^■9.  Divide  27a''62c2  by  9a be. 

^HO.  Divide  64a-^y^z^  by  Z'Zay^z', 

^Bl.  Divide  S8a*6«c8  by  lla-^6V. 

43.  It  follows  from   the  preceding  rub,  that  the  exact 
division  of  monomials  will  be  impossible, 
^B  1st.  When  the  co-efficient  of  the  dividend  is  not  exactly 
divisible  by  that  of  the  divisor. 

2d.  When  the  exponent  of  the  same  letter  is  greater  in 
e  divisor  than  in  the  dividend. 

3d.   When  the  divisor  contains  one  or  more  letters  not 
found  in  the  dividend. 
^H  When  either  of  these  three  cases  occurs,  the  quotient  may 
^^e  expressed  under  the  form  of  a  monomial  fraction  ;  that 
b,  a  monomial    expression,   necessarily   affected   with    the 
^Klgebraic  sign  of  division.      Such  expressions  are  said   to 
^^be  in  their  simplest  form^  when  the  numerator  ^nd  denomi- 
nator do  not  contain  a  common  factor. 

For  example,   12a^6-c(/,  divided  by   Sa^fic^,  gives 
12«^6W. 
8a26c2    ' 


43.  What  is  the  first  case  named  in  which  the  di vision  of  monomials 
«rill  not  be  exact  ?  What  is  the  second  ?  Wliat  is  the  third  ?  If  either 
of  these  cases  i)ccur,  can  the  exact  division  l)e  made  ?  Under  what  forn* 
iviil  the  quotient  then  remain  I  May  this  fraction  be  often  reduced  to  u 
uimplor  form  I 


64  ELEMENTARY     ALGEBRA. 

which  may  be  reduced  by  dividing  the  numerator  and  deno- 
minator by  the  common  factors  4,  a^^  b,  and  c,  giving 

I2a^b^cd         Za^bd 


Also, 


8a26c2  2c 

25a5i2^  5a 


l^a%H^         ZbH 


44.  Hence,  for  the  reduction  of  a  monomial  fraction  to 
its  simplest  form,  we  have  the  following 

RULE. 

Suppress  every  factor,  whether  numerical  or  literal^  that  is 
comm.on  to  both  terms  of  the  fraction,  and  the  result  will  be 
Hie  reduced  fraction  sought. 

From  this  new  rule  we  find, 

(1)  (2) 

ASa^^cd^    _  W.    ^^^   37a  bh^d    _  Zlb^e  . 
Z^a^Wde  ~  36^*    ^^        ^a^b  cH^  ~    QaH' 

(3)  (4) 

also      _2^  =  ±:    and    ^-^^^  ^ 


14a362         2ab  %a¥  W^ 

Ifbc 


5.  Divide   ^da^^c^   by    14a36c*.  Ans. 

6.  Divide   Gamn   by   Sabc.  Ans. 

7.  Divide    ISa^'^mn^   by    I2a'^b^cd.  Ans. 


2a 
2mn 

Zmn^ 


2a^b^cd 


44.  Give  the  rule  for  the  reduction  of  a  monomial  fractioa 


I 


G5 

.  0 

'         d 

25x1/ 

,2.  Divide   Sbni^nHx^u^   by    l^am^nf.       Ans.    —- — r^— • 

127 
V^dx^y'^ ' 


DIVISION. 

8.  Divide   28aVA-"'(f'  by    ICaiVrm.  ^ns. 

9.  Divide    I'Za^'cU-'  by    i2a^c*b'd. 

10.  Divide  lOOa^^xmn  by  25a36*rf.  ^//5 

11.  Divide   dGa^b^c^df  by   TSa^cary.  ^ns. 


3.  Divide  l27d^xY    by    16t/*x*y*. 
45.  If  we  have  expressions  of  the  form 


Ans. 


a        a'      a" 


&c., 


and  apply  the  rule  for  the  exponents,  we  shall  have 


,2-2  _  „0 


a^,  &c. 


It  since  any  quantity  divided  by  itself  is  equal  to  1,  it 
llows  that 


—  =  ao  =  1,    %-  =  a2-2  =  a«  =  1,  &c., 


finally,  if  we  designate  the  exponent  by  wi,  we  have 

a"* 

—  =  a"*-"*  =  ao  =  1  ;  that  is, 

a" 

^ie  power  of  any  number  whose  expo7ieni  is  0,  is  equal  to  1  ; 
id  hence,  a  factor  of  the  fljrni  a"  may  be  omitted,  being 
lual  tu  1. 

45.   What  is  a»  equal  to  ?     What  is  6«  equal  to  ?     What  is  the  poA/er 

I  CUV  number  equal  tn,  Avlicii  the  exjMiiient  of  the  jxjwer  is  01 


(UJ  KLKMENTAUY     ALGKBUA. 

2.  Divide  QaWc^d  by  2o?hH. 

^az-lP-d 

3.  Divide  %a^hhH^  by  4a^-Pc*d^.  Ans,  2^'-^. 

4.  Divide  16a6Z.«<;9  by  Sa^^^c^.  ^?^s.  2^^. 
6.  Divide  S2m^n^x'^7/'^  by  4m^w.3:ry.  Ans.  Sxy 
0.  Divide  OGa^fiSc^ScQ  ^y  24a*54(/5c9.  ^ns.  46(^3 

SIGNS  IN  DIVISION. 

46.  The  object  of  division,  is  to  find  a  third  quantity 
called  the  quotient,  which,  multiplied  by  the  divisor,  shall 
produce  the  dividend. 

Since,  in  multiplication,  the  product  of  two  terms  having 
the  same  sign  is  affected  with  the  sign  +,  and  the  product 
of  two  terms  having  contrary  signs  is  affected  with  the 
sign  — ,  we  may  conclude, 

1st.  That  when  the  term  of  the  dividend  has  the  sign  -}-, 
and  that  of  the  divisor  the  sign  of  +,  the  corresponding 
term  of  the  quotient  must  have  the  sign  +. 

2d.  When  the  term  of  the  dividend  has  the  sign  +,  and 
that  of  the  divisor  the  sign  — ,  the  corresponding  term  of 
the  quotient  must  have  the  sign  — ,  because  it  is  only  the 
sign  — ,  which,  multiplied  with  the  sign  — ,  can  produce 
the  sign  +  of  the  dividend. 

46.  What  will  the  quotient,  multiplied  by  the  divisor,  be  equal  to  I 
If  the  multiplicand  and  multiplier  have  like  signs,  what  will  be  the  sign 
of  the  product  ?  If  they  have  contrary  signs,  what  will  be  the  sign  of 
the  product  ?  When  a  term  of  the  dividend  and  the  term  of  the  divisof 
have  the  same  sign,  what  will  be  the  sign  of  the  corresponding  terra  of 
ilie  quotient  *  When  they  have  different  signs,  what  will  be  the  sign  of 
the  term  of  tije  quoticut  f 


D I  V  IK  I O  K 


07 


3d.  When  the  term  of  the  dividend  has  the  sign  —,  and 
that  of  the  divisor  the  sign  -)-,  the  term  of  the  quotient 
must  have  the  sign  — .     Again  we  say  for  brevity,  that, 


L 


4-  divided  by  4",  and  —  divided  by 
—  divided  by  +,  and  -f  divided  by 


give  4- 
give  -. 


EXAMPLES. 


1.  Diviie  4tax  by  —  2a.  Ana.  —  22L 

Here  it  is  plain  that  the  answer  must  be  —  2ar ;  for, 
^y  —  2a  X  —  2ar  =  +  4aar,  the  dividend. 


2.  Divide  SQa^x^  by  —  12a^x. 

3.  Divide  -  bSa^b''c^(P  by  2da^*c. 

4  Divide  -  84a*65d3  by   -  42a^^d. 

5  Divide  G4c*d^x^  by  IQc^dx. 

6.  Divide  -S8b*x^y^  by  —24Pcdx^.        A71S.  + 

17.  Divide  77aV^  by  -Ua^fz*. 

8.  Divide  84a-»6Vi  by  -  42a'b^c^d. 


.9.  Divide  —  60a'b^c*d  by   —  12a8rc'*(£2, 


0. 
11. 
12. 

i 

18. 


Divide  — 88a«6V  by  Sa^b^^c^ 
Divide  10^2  by   —  8;r. 
Divide  —  I5a^xj/^  by  3ay. 
Divide   —  84a63x  by   —12^2. 
Divide   —  d(]a*b^c^  by   I2a^c, 
Divide   —  144a9iVt/s  by  —  SGa*b^c^d, 
Divide  25GaVyc2x3  by    —  l()a2c2-2. 
Divide   —  300a*6^c3^2  by  30a*6V^. 
Divide  500a»6»c«  by   -  lOOct'iV. 


^n5.  —  3aa; 

Ans.  —2ahcd^ 

Ans.  2a^bH'^ 

Ans.  4d*x^ 

ny 

3cc^ 

^W5.   —  7 

Ans.  —  2 

abed 

Ans.  —  lla6. 

Ans.  —  2x. 

Ans.  —baxy"^. 

Ans.  7abx. 

Ans.  —  Sabc^. 

Ans.  4a^b-cd*. 

Ans.  —  ](jaf)cx 

Ans.  —  lldhcx 

Ans.  —  'Milte^ 


\a  —  X 


a  —  X 


68  KLEMENTAKY     ALOEBKA. 

19.  Divide    —  64a5Z/V    by    —  '^al^JPc^.  Ans.  Sabc. 

20.  Divide    +  dQa^b^d^  by    —  24:aHM.    Ans,  —  4ab'^(P, 
31.  Divide   72a^b^d'^   by    —  Sa^b^d.  Ans,  —  dabd^. 

Division  of  Polynomials. 

FIRST   EXAMPLE. 

47.  Divide   a^  —  2ax  -\-  x^   by    a  —  x. 

It  is  found  most  convenient,  Dividend.       Divisor, 

in  division  in  algebra,  to  place  a^  —  2ax  +  ^^ 

the  divisor  on  the  right  of  the  a^  —    ax 
dividend,  and  the  quotient  di-  —    ax  -^  x^, 

rectly  under  the  divisor.  —    ax -\-  x^ 

We  first  divide  the  term  a^,  of  the  dividend,  by  the  term 
u  of  the  divisor :  the  partial  quotient  is  a,  which  we  place 
under  the  divisor.  We  then  multiply  the  divisor  by  a,  and 
subtract  the  product  a^  —  ax  from  the  dividend,  and  to  the 
remainder  bring  down  x^.  We  then  divide  the  first  term 
of  the  remainder,  —  ax  by  a,  the  quotient  is  —  x.  We 
then  multiply  the  divisor  by  —  a;,  and,  subtracting  as  before, 
we  find  nothing  remains.    Hence,  a  —  x\%  the  exact  quotient. 

In  this  example,  we  have  written  the  terms  of  the  dividend 
and  divisor  in  such  a  manner  that  the  exponents  of  one  of  the 
letters  go  on  diminishing  from  left  to  right.  This  is  what  is 
called  arranging  the  dividend  and  divisor  with  reference  to 
that  letter.  By  this  preparation,  the  first  terra  on  the  left 
of  the  dividend  is  the  one  which  must  be  divided  by  the 
first  on  the  left  of  the  divisor,  in  order  to  obtain  the  first 
term  of  the  quotient. 

47 •  What  do  you  understand  by  arranging  a  polynomial  with  refer 
ouae  to  a  particular  l^t-ter  ? 


UIVI 3I0N 


(w 


48    Hence,  for  the  division  of  polynomials,  we  have  this 


RULE. 


I;.  Arrange  the  dividend  and  divisor  with  reference  to  the 
le  letter^  and  then  divide  the  first  term  on  the  left  of  the 
idend  by  the  first  term  on  the  left  of  the  divisor^  the  result 
he  first  term  of  the  quotient ;  multiply  the  divisor  by  this 
/I,  and  subtract  the  j^roduct  from  the  dividend. 
11.    Then  divide  the  first  term  of  the  remainder  by  the  first 
of  Vie  divisor^  which  gives  the  second  term  of  the  quotient ; 
\ltiply  the  divisor  by  the  second  term^  and  subtract  the  pro- 
:tfrom  the  result  of  the  first  operation     Continue  the  same 
:ess  until  you  obtain  0  for  a  remainder^  or  until  the  first 
of  the  remainder  cannot  be  divided  by  the  first  term  of 
divisor, 

8BC0NP   BXAMPLS. 

pt  it  be  required  to  divide 
h\a^^  -f-  10a*  —  4Sa^b  —  lob^ -{  4ab^  by  4ab  -  ba^-\-Sb^. 
le  here  arrange  with  refcrenc«  to  a. 
Dividend. 


10a*-4Sa36-)-  51a262+  Aa¥ 

^  10a<—    8</36  >-•   6a^6^ 

^^0a36T~57a262-}-  4ah^ 


15^< 


i5i* 


k 


—  AOa^b  -f  3'^a262-f-  2\ab^ 


Divisor, 
-  5a2  -f  4a6  -f-  36» 

Quotient, 


25a262_20a63-156* 
25a262~.20a63_l56* 


48.  Give  the  general  rule  for  the  division  of  polyrK-'nrJtU  I     If  the 

t  term  of  the  arranged  dividend  is  not  divisible  by  tht^  frat  t^^rm  of 

tlie  arranged  divisor,  is  the  exact  division  possible?     If  tli*»  6r*t  {t-nn  of 

any  partial  dividend  is  not  divisible  by  he  first  term  of  ihe  sSiv^r.  i* 

the  exact  divisioo  possible  f 


70 


ELEMENTARY     ALGEBRA 


Hemark. — When  the  first  term  of  the  arranged  dividend 
is  not  exactly  divisible  by  that  of  the  arranged  divisor,  the 
complete  division  is  impossible ;  that  is  to  say,  there  is  not 
a  polynomial  w^ich,  multiplied  by  the  divisor,  will  produce 
the  dividend.  And  in  general,  we  shall  find  that  the  exact 
division  is  impossible,  when  the  first  term  of  any  one  of  the 
partial  dividends  is  not  divisible  by  the  first  term  of  the 
divisor. 

GENERAL    EXAMPLES. 

1.  Divide  \Sx^  by  9x.  Ans.  2ar. 

2.  Divide  lO^-^y^  by  —  6x^i/.  Ans.  —  2y. 

3.  Divide  —  dax'^i/^  by  Qx'^y.  Ans.   —  ay, 

4.  Divide  —  %x'^  by   —  2;c.  Am.   +  4^:. 

5.  Divide  10a6  +  15ac  by  5a.  Ans.  26  +  3c. 

6.  Divide  30aa;  —  54:r  by  ^x.  Ans.  da  —  9. 

7.  Divide  lOar^y  —  15y2  _  5y  by  5?/.  A?is.  2x^  —  Si/  —  l. 

8.  Divide  12a  +  Sax  —  18ax^  by  3a.  Ans.  4  -f  a;  —  Gx\ 

9.  Divide  Gax^  +  9a^x  +  o?^^  by  ax.  Ans.  6a;  -f  9a  +  ax. 

10.  Divide  a^  -h  2aa;  +  a;^  by  a -\-  x.  Ans.  a -{-  x. 

11.  Divide  e,^  —  Sa^y  +  Say^  —  y^  hy  a  —  y. 

A71S.  a^  —  2ay  +  y\ 

12.  Divide  24a26  —  12a3c62  —  Cmb  by  —  Qab. 

Ans.  —  4a  +  2a'^cb  -J-  1. 

13.  Divide  6x*  —  96  by  Sx  —  6.  Ans.2x^-i-4x^+Sx-]-lQ. 

14.  Divide  .     .     a^  -  5a*a;  +  lOa^a;^  -  lOa^x^  +  ^ax*  — 
p6  by  ^2  —  2aa;  +  x\  Ans.  a^  —  Sa'^x  +  3aa;2  —  x\ 

15.  Divide  48a;3  —  7(jax^  -  64a2x  -f  105a3  by    2x  —  3a, 

Ans.  24x^  —  2ax'-  S^a\ 


DIVISION. 


71 


16.  Divide   y«  —  3y*x2  +  Sy'^x*  —  ar«  by   y^  —  'Sfz  + 
5yx2  —  x\  Ans.  y^  +  Sy^a:  +  3yz2  -|-  ar^. 

17.  Divide   64a*6«  —  25a^68   by   Sa^ft^  +  5a6*. 

18.  Divide   Ga^  +  2Za%  +  22aZ.2  ^  5^3    \^y   3^2  .^  4^5 
4  ^»2.  ^ws.  2a  4-  56. 

19.  Divide   Qcuxfi  +  CcltV  -f  ^^aH'^   by   ax  +  5aa:. 

Ana.  x^  -f-  ary®  -f-  lax, 

20.  Divide  -  15a*  +  STa^^c/  _  20aV/-  'ZOU^-d"-  -f-  446f(f/ 
-  8cy2   by    3a2  -  bid  +  cf.         Ans.  -  5a^  +  4//(i  -  8c/. 

21.  Divide   x*  -\-  x'^tp-  +  y*    by   x^  —  ary  +  y^. 

Ans.  x^  4-  ary  +  y'. 

22.  Divide  a:*  —  y*  by  ar  —  y.  Ans.  x^  +  ^V  "f"  ^V^  ~^"  y^* 

23.  Divide  3a*  -  Sa^fiz  -f  30^  4-  56*  -  36V  by  a^  -IP-. 

Ans.  3a2  -  56^  +  3c^ 

24.  Divide  Gj:«-5j-*y^  -GxV  +  Gary  +  15a:V  —  9a;V 
-f-  lOarV  +  15y«  by  3x'  -f  2^^^  +  3y^ 

-4;w.  2a;3  —  3a:2y2  4.  5^3^ 

25.  Divide  —  c^  +  IGa^ar'  —  7a6c  —  4a26x  —  ^a^V^^  Gacaj 
by    8ax  —  Ga6  —  c.  Ans.  2aa;  ■\-  ah  -\-  c, 

2G.  Divide  .    .  3ar*  +  4a;3y  -  4a:2  —  4a:V  4-  IGa-y  —  15 
f-y   2a-y  -f-  ar2  —  3.  Ans.  3a;2  —  2a:y  +  5. 

27.  Divide   ar'  -h  32y*   by   ar  +  2y. 

Ans.  X*  —  2.r>  +  4a'V  —  ^a^^  -f-  lOy*. 

2' .  Divide  3a*  —  2Ca3i  —  Hai^  4.  37a262    by    2ir2  - 
'ki6   ^  3a'.  Ans.  a*  —  7ai. 


72  ELEMENTARY     ALGEBRA, 


CHAPTER  II. 

Algebraic  Fractions, 

49.  Algebraic  fractions  are  of  the  same  nature  aj  irith- 
metical  fractions ;  that  is,  we  must  conceive  that  sonns  unit 
one  has  been  divided  into  as  many  equal  parts  as  there  are 
units  in  the  denominator,  and  that  one  of  these  parts  is 
taken  as  many  times  as  there  are  units  in  the  numerator. 
Hence,  addition,  subtraction,  multiplication,  and  division, 
are  performed  according  to  the  rules  established  for  arith- 
metical fractions. 

It  will  not,  therefore,  be  necessary  to  demonstrate  these 
rules,  and  in  their  application  we  must  follow  the  procedures 
indicated  for  the  operations  on  entire  algebraic  quantities. 

50.  Every  quantity  which  is  not  expressed  under  a  frac- 
tional form  is  called  an  entire  quantity. 

61.  An  algebraic  expression  which  is  partly  entire  and 
partly  fractional,  is  called  a  mixed  quantity. 


49.  How  are  algebraic  fractions  to  be  considered  ?  What  doe«  the 
denominator  show  ?  "What  does  the  numerator  show  f  How  then  ar* 
the  operations  in  fractions  to  be  peiformed  ? 

50.  What  is  an  entire  quantity? 

51.  What  is  a  mixed  quantity  I 


ALGEBRAIC     FRACTIONS.  78 

CASE    I. 

To  reduce  a  fraction  to  its  simplest  form. 

52.  A  fraction  is  said  to  be  in  its  simplest  form,  when 
there  is  no  common  factor  in  the  numerator  and  denomi- 
nator. The  rule  for  reducing  a  monomial  fraction  to  its 
fcimplest  form  has  already  been  given  (Art.  44). 

With  respect  to  polynomial  fractions,  examples  under  th« 
following  cases  are  easily  reduced. 

1.  Take,  for  example,  the  expression 

a2  -  2ab  +  62  * 

This  fraction  can  take  the  form 

(g  +  ^)  {<i  -  b) 
(«  -  by 

(Art.   89   and  40).     Suppressing  the  factor   a  —  6,  which 
is  common  to  both  terms,  we  obtain 

g  -i-  6 
a  —  b 

2.  Again,  take  the  expression 

5a3  —  lOa^b  -f  5ah* 
8a3  -  8a26 

This  expression  can  be  decomposed  thus: 

5a(a2  _  2q6  -h  6^) 
Sa\a  —  b)  * 

5a  (g  -  by 
®''  ea\a-b)    ' 


b9t  How  do  jou  reduce  a  fraction  to  its  simplest  terms  ? 
4 


74  ELEMENTARY     ALGEBRA. 

Suppressing  the  common  factors  a{a  —  6),  the  result  is 

5(a  -  b) 

Sa 

Hence,  to  reduce  any  fraction  to  its  simplest  form^  we  s^ip- 
press  or  cancel  every  factor  common  to  the  numerator  and 
ienominator 

Note. — Find  the  common  factors  of  the  numerator  and 
Ienominator  as  explained  in  (Art.  41). 

EXAMPLES. 

1.  Reduce  -— — ^  ,  „   to  its  simplest  form. 

4a2  +  2ac2 

2.  Reduce—-     0-7— qao""  ^^         simplest  form. 

Ans,     Y\ 

3.  Reduce  g    ■     to  its  simplest  form. 


Ans. 


17 


4.  Reduce   • ^   to  its  simplest  form. 


3c' 


Ans. 


5c 


^  „  ^     27a46*  —  81a  6«         ^    Sa^  -  962 
5.  Reduce  -— — — ^^  „,.  »       -aw*. 


63a  66  _  36a26*  *    76^  -  4a  * 

(5    Reduce ■  to  its  simplest  form.     Ans.  —  8. 

—  \2aWc 

246*  —  36a6*  .  46  —  Qa 

^'   ^^^^c®   48a*6*~CGa^  "  '*  8a*  -  llafi6^ 


ALGKURAIC     F  ft  ^^^10  VS.  75 

CASE  n. 
63.  To  reduce  a  mixed  quantity  to  the  form  of  a  frao 

UOD. 

RULE. 

Multif.hj  the  entire  part  by  the  denominator  of  the  fraction  ; 
idd  to  this  product  the  numerator^  and  under  the  result  plact 
the  given  denominator, 

EXAMPLES. 

1.  Reduce  6|  to  the  form  of  a  fraction. 

43 

6x7  =  42:     424-1=43:     hence,     6}=^ 

2.  Reduce  x  — ~  to  the  form  of  a  fraction. 

X 

a2  -x^        x^-  (a^  -  x^)       2x2  __  ^2 
z =  ^^ = Ans, 

XXX 

ax  "4~  X 

3.  Reduce  x ;; to  the  form  of  a  fraction. 

2a 

ax  —^ 
Ans. 


2a 


2x  —  1 

4.  Reduce  5  -\ r to  the  form  of  a  fraction. 

ox 

,       17* -7 

X  ~~'  a  ~~-  \ 

6.  Reduce  1 to  the  form  of  a  fraction. 

a 

2a  —  a;  -f  1 
Ans. • 


&3   Ilivw  do  you  roduoo  a  mlxwl  quantity  to  tbo  fumi  of  i  fraction  t 


76  ELEMENTARY     ALGEBRA. 


^ 3 

6.  Reduce     1  +  2a: —     to  the  form  of  a  fraction. 

5x 


bx 

7.  Reduce     2a-\-b to  the  form  of  a  fraction, 

o 

16a  +  8i-3c— 4 
Ans. 


QcL^x  —  ah 

8.  Reduce  Qax  -f  6 ;; to  the  form  of  a  fraction, 

4a 

\^a?x  4-  5a5 
Ans, 


4a 

8  _L.  Qa^Px* 
9.  Reduce  8  +  oab To^  T~i —  ^^  *^®  ^^^'^  of  a  fraction 

mabx^  -h  30a262a:*  -  8 


12tt6a;* 
352  _  8^4 

10.  Reduce  9  4- rr-  to  the  form  of  a  fraction. 

a  —  0^ 

,       9a  —  662  _  8c* 

Ans. 

a  —  0^ 


CASK  in. 
54.  To  reduce  a  fraction  to  an  entire  or  mixed  quantity. 

RULE. 

Apply  the  process  for  division  until  the  first  term  of  the 
remainder  is  not  divisible  by  the  first  term  of  the  divisor.  To 
the  quotient^  thus  obtained,  add  the  last  remainder  divided  by 
the  denominator. 

54.  Ho-w  do  you  reduce  a  fraction  to  an  entire  or  mixed  quantity  f 


ALOKBRAIC     FRACTIONS.  T7 


EXAMPLES. 


8066 
i.    lieduce    — - —    to   an    entire   number, 
o 

8)8966( 


1120  ...  6  rem. 

Hence,  11205  =  ^^' 


ax  —  a 


2.  Reduce    to   a   mixed   quantity. 

Ajis.  a . 


ax  —  X 


3.  Reduce    to  an  entire  or  mixed  quantity. 


Ans.  a  —  X, 


4.  Reduce    r to  a  mixed  quantity. 


Ans.  a 

b   ' 

a}  —  x^ 

5.  Reduce    to  an  entire  quantity.     Ans.  a -h  x, 

a  —  X 


6.  Reduce     ?_   to  an  entire  quantity. 

Ans.  x^  -\-  XT/  -\-  y' 

„    ^  ^           l0x^-5x  +  2  .     ^ 

,7.  lieduce to  a  mixed  quantity. 

DX 

A718   2x  —  I  -\-  —. 
bx 

„    ^  J         S6;r3  -  72aj  +  S2a^x^  .     ^ 

8.  Reduce to  a  mixed  quantity. 


78  ELEMENTARY     ALGEBRA 


CASE    IV. 

55.  Tc  reduce  fractions  having  different  denominators  to 
equivalent  fractions  having  a  common  denominator. 


RULE. 

Mulliply  each  numerator  into  all  the  denominators  except 
its  own,  for  new  numerators,  and  all  the  denominators  iogethei 
for  a  common  denominator, 

EXAMPLES. 

1.  Reduce  J,  J,  and  f ,  to  a  common  denominator. 

1x3x5  =  15  the  new  numerator  of  the  1st. 
7  X  2  X  5  =  70  "  "         "         2d. 

4  X  3  X  2  =  24  "  "         "        3d. 

and     2  X  3  X  5  =  30  the  common  denominator. 

Therefore,   J^,   |g,   and  JJ,  are  the  equivalent  fractions. 

Note. — It  is  plain  that  this  reduction  does  not  alter  the 
values  of  the  several  fractions,  since  the  numerator  and 
denominator,  of  each,  are  multiplied  by  the  same  number. 

2.  Reduce    -7-    and    —    to  equivalent  fractions  having 

0  c 

a  common  denominator. 

ax  c  =  ac  i^^^  ^^^  numerators. 
6  X  6  =  6M 
and  bx  c  =  hc      the  common  denominator. 

DO*  II JW  do  you  roduco  fractioiis  to  a  coinmou  dcnoiuiiiaUir  t 


ALQKBRAIO     FRACTIONS.  70 

Hence,  —  and   —   are  the  equivalent  fractions. 

3k  Reduce  —    and    to  fractions  having  a  coin. 

0  c 

Dion  denominator.  Ans.  -r-    and     ; • 

be  be 

4.  Reduce      —- >     — -?     and    d.    to   fractions    havinflj    a 

,  .  .         9cx      4ab  ,     C)acd 

ooninion  denominator.  Ans.  — — »    - — >  and  • 

hac      Gac  (iac 

3  2x  2x 

5.  Reduce  -— >     -^r-,    and    a  H ,  to  fractions   having 

4  o  a 

a  common  denominator. 

9a      Sax         ^    12«2  -|-  21a; 

^7M.  >  i  and   • 

12a       12a  12a 

1  a^  a2  _j_  ^2 

6.  Reduce     --,     -— >     and     »     to  fractions  hav- 

2  3  a  +  X 


ing  a  common  denominator. 


3a  +  3x     2a^-\-2a^x        ^    6a^ -\- 6x^ 
Ans,  - — -— r»    p; -— — ,  and 


6a  -{-Qx    (ja   +(jx    '  6a   +  6* 

a      (jax  a,^  —  x^ 

7.  Reduce  — rj  —z—i    and     ; — >     to    a    common 

36      5c  a 

denominator. 

.          bacd        ISabdx           ,      ISa^k  —  I5bcx* 
^'^'-   154^'     l5W'     ^"^     156^5 • 

«    T^    -1  c       a  —  b  .  c 

8.  Reduce     -->     j     and     ->     to   a   common 

5a  c  a  4-  6 

deixDminator. 

.  ac^-{-c^  5a3-5a62  5ac» 

Ans,  — - — .  ,  .  )    — ^; — .  ,     ->    and 


Sa^c  -I-  5a6c      5a2c  -f  5a6c  5a^c  -f-  5a^ 


80  ELEMENTARY  ALGEBRA 

CASE  V. 

56.  To  add  fractional  quantities, 

RULE. 

Reduce  the  fractions,  if  necessary,  to  a  common  denomtrui' 
tor ;  then  add  the  numerators  together,  and  place  their  sum 
over  the  common  denominator, 

EXAMPLES. 

1.  Add  f,  I,  and  |  together. 

By  reducing  to  a  common  denominator,  we  have 

6x3x5  =  90  1st  numerator. 

4  X  2  X  5  =  40  2d  numerator. 

2X3X2  =  12  3d  numerator. 

2  X  3  X  5  =  30  the  denominator. 

Hence,  the  expression  for  the  sum  of  the  fractions  becomes 

90       40       12  _  142  . 
30  "*"  30  "^  30  ~  "30" ' 

which,  being  reduced  to  the  simplest  form  gives  4^. 

2.  Find  the  sum  of  — ,    — ,    and   — ;• 

0      a  f 

Here     a  x  d  x  f=  adf  ■\ 

c  X  h  X  f=  chf  \  the  new  numerators. 

t  Xh  Xd  —  ebd  ) 
And      h  X  d  xf=  bdf     the  common  denominator. 
_  adf   ,     chf    .     ehd        adf  -\-  chf  +  ^^d 

^''"^'  w+w  +  w  =  -^3-f — ' '''  ^"^ 

G6.  How  do  you  add  fractions ! 


ALGEBRAIC     FRACTIONB.  81 


3.  To  «-^    add    4  +  ^ 
6  c 


2abx  -  3cx» 

A?is.  a  4-  6  4  i 

be 

X 


4.  Add    -— ,     — -     and     -—    together.  Ans.  -\-  —^ 

2         3  4  12 

IT     Aj^      ar  — 2        ^    4x  ,  ,        19a:— 14 

5.  Add     — - —  and    —   together.         Ans,    — - 


7        °  21 

p    \aa        1^-2.    o      ,2a:-3       .         .     ,   102r-17 

6.  Add  X  -\ —  to  2x  -\ : — .  Atis.  4x-\ — . 

3  4  12 

5x^  X  I   ct 

7.  It  is  required  to  add  4ar,  -^'  and  — - —  together. 

.        .         5ar3 -f  ax -j- a2 
Afis.  4a;  -f  — 


2ax 
8.  it  is  required  to  add  — ,  — ->  and   — - —    together. 

.        o     .      49a;+12 
Ans.  Zx  -\- 


1 X  X 

9.  It  is  required  to  add   4ar,  — ,  and  2  -f-  —   together. 


60 

9 5"   '" 


^        ^      ,    44a:-f-90 

Ans.  4a:  -\ — • 

4o 

2x                    8a: 
10.  It  is  required  to  add  Zx  -\-—   and  x —  together. 

o  «/ 

23a: 
Ans.  3a:  -f 


45 

11.  Required  the  sum  of  oc  —  — -    and    1 -. 

8a  c? 

%a?cd  —  Gbd  -^  Sad  — Sac 

^'"- sild 

4* 


O*  ELEMENTARY  ALGEBRA. 

CASE  VI. 

57    To  subtract  one  fractional  quantity  from  another. 

RULE. 

I.  Reduce  the  fractions  to  a  common  denominator. 

II.  Subtract  the  numerator  of  the  suhtrahend  from  tht 
numerator  of  the  minuend^  and  place  the  difference  over  the 
tommon  detiominator, 

EXAMPLES. 

3  2 

1.  What  is  the  difference  between   -— •  and  — -• 

7  o 

y  ~  "S"   ""  56        56  ~  50  "~  28  *      ^*' 


2.  Find  the  difference  of  the  fractions      ^,      and  — •• 

2o  oc 

{    (x—    a)  Xoc  =  OCX  —  Sac  )    , 
Ilere,-^,^         .  (       „,        ,   ,       ^i    [  the  numerators. 
'  (  (2a  —  4x)  X  26  =  4a6  —  862;  ) 

And,  26  X  3c  =  66c   the  common  denominator, 

3ca:— Sac       4a6  — 86a;       3ca;— Sac— 4a6-f86a;      . 

"<="<=«' -6k 66r-  = Wc •  ^"* 

,      ,.^  .  12a;       ,  Zx  ,        39a; 

3.  Required  the  difference  of  -rr-  and  — -  •        Ans.  — - 

3v  37v 

4.  Required  the  difference  of  5y  and    — .       ^715.     -^ 

3a;        _   23;  .         13.2; 

5.  Required  the  difference  of   -=-  ^-nd  —  •       Aiis.  — -- 


67.  How  do  you  subtract  fraclbns  I 


ALOEBHAIO     FRAOTIONS.  83 

X    I     CL  C 

6.  Required  the  difference  between    — r —  and  — . 

.       dx  •\-  ad  —  be 

Ans. j-z • 

bd 

7.  Required  the  difference  of  — -7 —    and    — - — • 

24ar  +  8a  — 10^»a:— 355 


Ans. 


406 


8.  Required  the  difference  of   Zx  -\-  -r-     and  a; • 

be 

A       ^        cx  -\-  bx  —  ah 

Ans,  2x  H -^ — 

be 

CASK  vn. 

58.  To  multiply  fractional  quantities  together. 

RULE. 
If  the  quantities  to  be  multiplied  are  mixed^  reduce  them  to 
fractional  forms  ;  then  multipli/  the  numerators  together  for 
a  numerator  and  the  denominators  together  for  a  denominator, 

EXAMPLES. 

1.  Multiply  i    of    i    by    8J. 

Operation. 
We  first  reduce  the  com- 
pound fraction  to  the  sim- 
ple one  j'lf,  and  then  the 
mixed  number  to  the  equiv- 
alent fraction  Y  ;  after 
which,  we  multiply  the 
numerators  and  denomina- 
tors together. 

42 


1 

0 

«'f"5. 

•i-l- 

Hence, 

3       25       75       25 
42^  3  ~12G~42 

84  ELEMENTARY     ALGEBSi 

2.  Multiply  a  H by  -r •    Jirst,   a-] = — 

a      "^    a      "^  a  a 

TT  a^  -\-  bx         c  a^c  +  bcx 

Hence,  x   —r  — -. Ans, 

a  a  ad 

«    T^       .     1    1  -,  ^  ox       ^   Sa  .         9a« 

3.  Kequired  the  product  of    —  and  — •  Ans.  -—p 


2a;  ^x^ 

4.  Required  the  product  of   —   and  -^ — 


3a:3 

Ans.    — — 

5a 

5.  Find  the  continued  product  of  — ?   - —   and  -— -  • 

^  a        c  26 

Ans.  9ax 

6.  it  is  required  to  find  the  product  of  6  H and 

Oi  X 


.          ah  -\-hx 
Ans.    


X 


a;2  _  52             a;2  +  52 
7.  Required  the  product  of  — 7 and  


^*  -  ^* 


62c  +  6c2 


a;  +  1  a:  —  1 

8.  Required  the  product  of  a:  + ?  and 


Ans, 


a  a  -\-  h 

ax^  ~  ax  -\-  x^  —  1 


a^  +  ab 
9.  Required  the  product  of  a  H by    —       ^  • 


Ans. 


2y2 


a 'a; 


aa;  +  ax^  —  x"^ 


&8<  How  do  you  multiply  fractious  together  ? 


ALGEBRAIC     FRACTIONS.  86 

CASK     VIII. 

69.  To   divide   one   fractional   quantity  by  another. 

RULE. 

Meduce  the  mixed  quantities,  if  any,  to  fractional  forms  ; 
tnen  invert  the  terms  of  the  divisor  and  multiply  as  in  the  last 
tase, 

EXAMPLES. 

1.  Divide.     .     .     •  24    ^^   "s" 

The  true  quotient  will  be    expressed  by   the  complex 

10 

fraction  ^« 
I 

Let  the  terms  of  this  fraction  be  now  multiplied  by  the 

denominator  with  its  terms  inverted :  this  will  not  alter  the 
value  of  thie  fraction ;  and  we  shall  then  have, 

T  =  -3^=^^=i?  X  f  =  f  =  quotient. 

It  will  be  seen  that  the  quotient  is  obtained  by  simply 
multiplying  the  numerator  by  the  denominator  with  its 
terms  inverted.  This  quotient  may  be  further  simplified  by 
dividing  by  the  common  factors  5  and  8,  giving  J  lor  the 
true  quotient. 


2.  Divide  .     , 

.     a 

-4 ''  f 

a 

6        lac-b 
2c-       2c 

6 
ence,  a  --  - 

3 

2ac~b       g 
-      2.       X/ 

B!).  How  «!o  yoii  divide  one  fracli*)n  by  auotl.cr? 


ELEMENTARY     ALOEBKA. 


8.  Let    — -    be  divided  by    -—  • 
5  -^      13 

4.  Let     -— -     be  divided  by     5x. 


5.  Let    — - —    be  divided  by    —  • 

o  o 

6.  Let    — '■ — -     be  divided  by     —  • 

X  1  li 

7.  Let    — -    be  divided  by     —  • 
^•^"'     ^     be  divided  by    ^ 
^•'^'    /-2bx  +  b'     be  divided  by    ~-^^ 


A718. 

60 

Am 

4:X 

^•35' 

'+1 

Ax 

Ans.  - 

2 
:-l 

Ans. 

6bx 
2a 

Ans.    • 

x-b 
Qc'x 

x^  +  bx 

Ans.  .  +  '-■ 
x 


10.  Divide   Ga^  +  --  by  c^  —   * 


5      •'  2 

60^2  +  2S 


Ans. 


10c2  _  5a;  4-  5o 


11.  Divide  18c2  -  a;  +  -^   by   a^  -  A 
6       •'  5 


90Jc2  —  5^a;  +  5a 


ba^b  -  62 


12,  Divide   20a;2  .-  M    oy  a:2  -  ^^ 


20c^cy22  -  86-5/ 

(/ry:c2  -  (Zc36"X^ 


KQUATiONtt     oy     TUK     laUtiT     UUUlCl^S.  87 


CHAPTER  m. 

0/  Equations  of  the  First  Degree, 

60.  An  Equation  is  the  algebraic  expression  of  two  equal 
quantities  with  the  sign  of  equality  placed  between  them. 
TnuSjX  z=  a-\-b  is  an  equation,  in  which  x  is  equal  to  the 
Bum  of  a  and  b. 

61.  By  the  definition,  every  equation  is  composed  of  two 
parts,  connected  by  the  sign  =.  The  part  on  the  left  of  the 
sign,  is  called  the  Jirst  member  ;  and  that  on  the  right,  the 
second  member.  Each  member  may  be  composed  of  one  or 
more  terms.  Thus,  in  the  equation  x  :=  a  -{-  b^  x  is  the  first 
member,  and  a  +  b  the  second. 

62.  Every  equation  may  be  regarded  as  the  algebraic 
enunciation  of  some  proposition.  Thus,  the  equation 
X  -\-  X  =  30,  is  the  algebraic  enunciation  of  the  following 
proposition : 

60i  "Wliat  is  an  equation  I 

01.  Of  how  many  parta  is  every  equation  composed  I  How  are  the 
parts  connected  with  each  other  ?  What  is  the  part  on  the  left  called  I 
What  is  the  part  on  the  right  called  ?  May  each  member  be  cornp<i»cd 
of  one  or  more  terms  ?  In  the  equation  x  =  a  +  6,  which  is  the  first 
uiciuber  t  Which  tlie  second  ?  Huw  many  terms  iu  the  firtit  member  I 
lluw  uiaay  m  t}ic  occotid  t 


88  ELKMENTARY     ALGEBRA* 

To  find  a  nurnber  which  being  added  to  itself,  shall  give  a 
sum  equal  to  30. 

Were  it  required  to  solve  this  problem,  we  should  first 
express  it  in  algebraic  language,  which  would  give  the 
equation 

a;  +  a;  =  80. 
By  adding  x  to  itself,  we  have 
2x  =  30. 
And  by  dividing  by  2,  we  obtain 
x  =  l5. 

Hence,  we  see  that  the  solution  of  a  problem,  by  algebra, 
consists  of  two  distinct  parts  :  viz.  the  statement  of  the 
problem,  and  the  solution  of  an  equation. 

I.  The  STATEMENT  covisists  in  expressing  algebraically  the 
relation  between  the  Jcnoivn  and  the  required  quantities. 

II.  The  SOLUTION  of  the  equation  consists  in  finding  the 
values  of  the  required  quantities  in  terms  of  those  which  are 
known. 

The  given  or  known  parts  of  a  problem,  are  represented 
either  by  figures  or  by  the  first  letters  of  the  alphabet,  a,  6, 
c,  &c.  The  required  or  unknown  parts  are  represented  by 
the  final  letters,  re,  y,  z,  &;c. 

EXAMPLE. 

Find  a  number  which,  being  added  to  twice  itself,  the 
sum  shall  be  equal  to  24. 

62.  How  may  you  regard  every  equation  ?  "What  proposition  does 
the  equation  a;  +  a;  =  30  state  ?  Of  how  many  parts  does  the  solution 
of  a  problem  by  algebra,  consist?  Name  them.  In  what  does  the  1st 
part  consist  ?  What  is  the  2d  part  ?  By  what  are  the  knov/n  parts  of 
"%  prouosition  represented  ?  By  what  are  the  unknown  parts  represeutccl  I 


KViUATIONa     OK     THE     FIKaX     UaORKJfi.  89 

Statement. 
Let  X  denote  the  number.     We  shall  then  have 

a-  +  2x  =  24. 
This  is  the  statement. 

Solution. 

Having     .     .     .     .  ar  +  2^  =  24, 

we  add x  +  2x, 

which  gives       ...  3a;  =  24, 

and  dividing  by  3,      .  x  =  S. 

63.  The  value  found  for  the  unknown  quantity  is  said  to 
be  verified^  when,  being  substituted  for  it,  in  the  given  equa- 
tion, the  two  members  are  proved  equal,  each  to  each. 

Thus,  in  the  last  equation  we  found  a:  =  8.  If  we  substi- 
tute this  value  of  x  in  the  equation 

a;  +  2j:  =  24, 
we  shall  have      8-|-2x8  =  8-fl0  =  24. 
which  proves  that  8  is  the  true  answer. 

64.  An  equation  involving  only  the  first  power  of  the 
unknown  quantity,  is  called  an  equation  of  the  first  degree. 
Tims,  6ar  +  3a:-5=  13, 

and  ax  -\-  bx  -\-  c  =i  d^ 

Are  equations  of  the  first  degree. 

By  considering  the  nature  of  an  eqjjation,  we  see  that  it 
must  possess  the  three  following  properties  : 


G3.  When  is  an  equation  said  to  be  terijied? 

64.  When  an  equation  involves  only  the  first  power  of  the  unknowu 
qnautity,  what  is  it  called  !  What  are  the  three  essential  propcrtiea  of 
every  oquatiuii  I 

& 


00  KLEMENTAUY     ALGEliUA 

1st.  The  t^v"^  members  must  be  composed  of  quantifies  of 
the  same  k.nd  :  that  is,  dollars  =  dollars,  pounds  =  pounds. 
2d.  The  two  members  must  be  equal  to  each  other. 
3d.  The  two  members  must  have  like  signs. 

65.  An  axiom  is  a  self-evident  truth.  We  may  lere 
state  the  following. 

1.  If  equal  quantities  be  added  to  both  meynbers  of  an  eqiia- 
tio7i,  the  equality  of  the  members  will  not  be  destroyed. 

2.  If  equal  quantities  be  subtracted  from  both  members  of 
an  equation^  the  equality  will  not  be  destroyed. 

3.  If  both  members  of  an  equation  be  multiplied  by  the.  same 
number f  the  equality  will  not  be  destroyed. 

4.  If  both  members  of  an  equation  be  divided  by  the  same 
number,  the  equality  will  not  be  destroyed. 

Transforraation  of  Equations. 

66.  The  transformation  of  an  equation  consists  in  chang 
ing  its  form  without  affecting  the  equality  of  its  members. 

The  following  transformations  are  of  continual  use  in  the 
resolution  of  equations. 

First  Transformation 

67.  When  some  of  the  terms  of  an  equatioi  are  frao. 
tional,  to  reduce  the  equation  to  one  in  which  the  terms 
shall  be  entire. 

I.  Take  the  equation 

2x      ^x       X 


65.  "What  is  an  axiom  ?     Name  the  four  axioms  ? 

66.  What  is  the  transformation  of  an  equation  ? 

67.  What  is  the  first  transformation?  What  is  tlje  least  •"mraou 
multiple  of  several  numbers?  How  do  you  find  the  least  cotmautf 
uiiiltiple  ? 


EQUATIONS     OF     THE     FIRST     DEOUKE.  D) 

First,  reduce  all  the  fractions  to  the  same  denominator, 
by  the  given  rule ;  the  equation  then  becomes 

72         72   "*■   72   ~       ' 

and  since  we  can  multiply  both  members  by  the  same  num 
ber  without  destroying  the  equality,  we  will  multiply  them 
by  72,  which  is  the  same  as  suppressing  the  denominator 
72,  in  the  fractional  terms,  and  multiplying  the  entire  term 
by  72 ;  the  equation  then  becomes 

4Sz  -  54x  +  12a?  =  792, 
or  dividing  by  6,    8j  —    9x -\-    2x  =  ]  32. 

But  this  last  equation  can  be  obtained  in  a  shorter  way,  b^ 
finding  theleast  conimon  multiple  of  the  denominators. 

The  lecuit  common  multiple  of  several  numbers  is  the  least 
number  which  they  will  separately  divide  without  a  remainder. 
When  the  numbers  are  small,  it  may  at  once  be  determined 
by  inspection.  The  manner  of  finding  the  least  common 
multiple  is  fully  shown  in  Arithmetic  §  87. 

Take  for  example,  the  last  equation 

3         4^0 

We  see  that  12  is  the  least  common  multiple  of  the  dt 
nominators,  and  if  we  multiply  all  the  terms  of  the  equa- 
tion by  12,  we  obtain 

8jr  -  9j-  -}-  2ir  =  132  * 

I  he  same  equation  as  before  foimd. 


02  ELEMENTAKY     ALGEBKA. 

68.  fTence,  to  make  the  denominators  disappear  from  uu 
equation,  we  have  the  followmg 

RULE. 

L  Find  the  least  common  multiple  of  all  the  denomtna- 
tors. 

II.  Multiply  every  term  of  both  members  of  the  equation  by 
this  common  multiple — reducing  at  the  same  time  the  frac- 
tional to  entire  terms, 

EXAMPLES. 


CC  X 

\.  Clear  the  equation   —  +  —-—4  =  3     of  its  denomi- 
nators. Ans.  Ix  -\-^x  —  140  =  105. 

XXX 

2.  Clear  the  equation   -^  -f-  -—  —  —-=:  8     of  its  denom- 

o        9       27 

inators.  Ans.  ^x  -\-  Q)X  —  2x  z=  432. 

X  X  X  X 

3.  Clear  the  equation    77  +  -^  —  77  +  77^  =  20      of  its 
denominators.  Ans.  18a:  +  \2x  —  4:X  -{- Zx  =  720. 

XXX 

4.  Clear  the  equation    —  +  ——-—  =  4     of  its  denom- 
inators.  Ans  \4.x  +  lOx  —  35^  =  280. 

XXX 

5.  Clear  the  equation -f-  —  =  15     of  itsdenom- 

4         5        6 

inators  Ans.  15.r  —  12a;  +  10^  =  900. 


(]S  Give  iLe  iiilu  for  cleariug  an  equation  oi  its  deiumiiiiatorB. 


ItQUATIONS     OF     THE     FIKST     DEORKK.  93 

6.  Clear  the  equation    — T"  +  "^  +  77'  —  ^'^    ^^  '^^ 

4       0       o       y 

deuominators.  Ans.  ISx  —  12 j:  -f  9x  +  8-c  =  804. 

a  c 

7.  Clear  the  equation    — -\-  f  =z  g. 

Ans.  ad  —  be  -\-  hdf  =  bdg, 

8.  In  the  equation 

ax       2c2x    ,    ,         4&c2ar       ba^   ,    2c^ 

—  4-  4a  =  — —  —  7T  H 36, 

6         ao  a^  b^         a 

the  least  common    multiple  of  the   denominators  is  a^i* ; 

hence,  clearing  the  equation  of  fractions,  we  obtain 

a^bx  —  2a'^bc^x  +  ^a*b^  —  46 Vx  —  5a«  -f  2«26V  -3a^63. 

Second  Transformation, 

69.  When  the  two  members  of  an  equation  are  entire 
polynomials,  to  transpose  certain  terms  from  one  member 
to  the  other. 

1.  Take  for  example  the  equation 

5ar  -  6  =  8  +  2x. 

If,  in  the  first  place,  we  subtract  2x  from  both  merabers, 
the  equality  will  not  be  destroyed,  and  we  have 

5x  —  G  —  2x  =  8. 

Whence  we  see,  that  the  term  2x,  which  was  additive  in 
the  second  member,  becomes  subtractive  by  passing  into 
the  first. 


69.  What  is  the  second  transformation  I  What  do  you  uiid<>r8tann 
by  transposing  a  term  ?  Give  the  rule  for  transposing  from  one  member 
tu  the  i)thor. 


M  ELEMENTARY     ALGEBRA. 

Ill  the  second  place,  if  we  add  6  to  both  members,  the 
ecjuality  will  still  exist,  and  we  have 

5^  -  6  -  2a;  4-  6  =  8  -f  6. 

Or,  since  —  6  and  +  6  destroy  each  other,  we  have 

Hence  the  term  which  was  subtractive  in  the  first  niem 
ber,  passes  into  the  second  member  with  the  sign  of 
addition. 

2.  Again,  take  the  equation 

ax  -\-  b  =  d  —  ex. 

If  we  add  ex,  to  both  members,  and  subtract  b  from 
each,  the  equation  becomes 

ax  -{-  b  -\-  ex  —  b  =  d  —  ex  -{-  ex  —  b. 

or  reducing  ax  -{-  ex  =z  d  —  b. 

When  a  term  is  taken  from  one  member  of  an  equation 
and  placed  in  the  other,  it  is  said  to  be  transposed. 

Therefore,  for  the  transposition  of  the  terms,  we  have  the 
following 

RULE. 

Any  term  of  an  equation  may  be  transposed  by  simjyly 
changing  its  sign  from  -\-  to  — ,  or  from  —  to  -\-. 

70.  We  will  now  apply  the  preceding  principles  t)  the 
resolution  of  equations. 

1.  Take  the  equation 

^U  ~  3  =  2a;  +  5. 


KQUA'/IONS     OF     THB     FIUUT     DEUIIEK.  95 

By  transposing  the  terms  —  3  and  2x,  it  becomes 
4c  —  2ar  =  5  -h  3. 
Or,  reducing  2j:  =  8. 

Q 

Dividing  by  2  ar  =  —  =  4. 

Verification, 

If  now,  4  be  substituted  in  the  place  of  ar,  in  the  giveu 
er  uation 

4j:  —  3  —  2j;  +  5, 
it  becomes  4x4  —  3  =  2x4  +  5. 

or,  13  =  13. 

Hence,  the  value  of  x  is  verified  by  substituting  it  for  the 
unknown  quantity  in  the  given  equation. 

2.  For  a  second  example,  take  the  equation 

5x      4.x       23  _  7        13.r 

12       3  ~  8  6    * 

By  causing  the  denominators  to  disappear,  we  have 

lOa?  -  32x  -  312  =  21  -  52ar, 
or,  by  transposing 

10a:  -  32a-  +  52a;  =  21  -f  312 
i>y  reducing  30a:  =  333 

333        111       ... 

"  =  ^  =  lo-  =  '^-'- 

a  result  which  may  be  verified  by  substituting  it  for  a?  in  the 
given  equation. 

3.  For  a  third  example  let  us  take  the  equation 

(3a  -x){a-b)  f  2aar  =  46  (a;  +  a). 


96  ELEMENTARY     ALGEBRA. 

It  is  first  necessary  to  perform  the  multiplications  indicat 
ed,  in  order  to  reduce  the  two  members  to  polynomials. 
This  step  is  necessary  before  we  can  disengage  the  unknown 
quantity  x^  from  the  known  quantities.  Having  done  that, 
the  equation  becomes, 

3a2  —  ax  —  Zah  -\-  hx  •\-  2ax  z=z  Aibx  -\-  4ab, 
or,  by  transposing 

—  ax  -]-  bx  -\-  2ax  —  4hx  =  4ab  +  3a6  —  Sa^, 
by  reducing  ax  —  Shx  =  lab  —  Sa^; 

Or,  (Art.  41).  (a  —  Sb)x  =  lab  -  Sa\ 

Dividing  both  members  by  a  —  36  we  find 
_  lab  —  Sa^ 
^-      a-'Sb  ' 

Hence,  in  order  to  resolve  an  equation  of  the  first  degre- 
we  have  the  following 

RULE. 

I.  If  there  are  any  denominators^  cause  them  to  disappean 
and  perform^  in  both  members^  all  the  algebraic  operation, 
indicated. 

II.  Then  transpose  all  the  terms  containing  the  unknown 
quantity  into  the  jirst  member^  and  all  the  known  terms  intc 
the  second  member. 

III.  Reduce  to  a  single  term  all  the  terms  involving  the  un 
known  quantity  :  this  term  vnll  be  com.posed  of  two  factors 
one  of  which  will  be  the  unknown  qvantity^  and  the  othe'^  itit 
multipliers,  connected  by  their  respective  signs. 

IV.  Divide  both  members  of  the  equation  by  the  multiplier 
of  the  unknown  quantity. 

70i  What  is  the  first  step  in  reserving  an  equation  of  the  first  degree 
What  the  second  ?     What  the  third  ?     What  the  fourth  ? 


KQUATI0N8     OF     THE     FIRST     DEUUKE.  9^ 


EXAMPLES. 

1    Given   3;j  —  2  +  24  =  31    to  find  x,  Ans.  ar  =  3. 

2.  Given   a;  +  18  =  3ar  —  5   to  find   x,      Ans.  a:  =  11  — ^ 

3.  Given   6  -  2i:  -f  10  =  20  -  3x  —  2   to  find    x. 

Ans.  X  =  2, 

4.  Given   x-\-— x-{-  —  x  =  \l    to  find  x,     Ans,  x  =  6 

2  o 

1  /* 

—  X  -\-  I  =bx  —  2, to  fmd  X.  Ans.  x  =  y 


5.  Given   2jr  —  —  a;  +  1  =  5x  —  2,to  find  x.  Ans.  ar  =  -;^ 


6.  Given    Sax  -\-  — 3  =  6x  —  a,  to  find   x. 

6  — 3a 


Ans.  X  = 


Qa-2b 


7.  Given  ^^  +  ^  =  20  -  ^-i^to  find  x. 


Ans.  a;  =  23  — 
4 


8.  Given  ^-^  -\- ^  =  4  -^-r-^  to  find  x. 

ti  o  4 


o6 

^7W.  X  =  3— 
lo 


/p       3x  4a; 

9.  Given  --  —  --  -|-  ar  =  —  —  3   to  find  x. 
4        2  o 


u4rw.  a;  =  4. 


10.  Given ; 4  =  /  tc  find  ar. 

€  a 


cdf  f  4c(/ 
6ad—2be 


98  ELEMKNTAAY     ALGEBRA. 

Sax  —  b       3b  — c        .       i    ^    n  j 

11.  (jriveii —  =4  —  0    to  find    x. 

7  2 

56  +  96  -  7e 
Ans.  X  = 


IQa 


lo    /-•         ^        X-  2   ^    X        13    ^     -    - 

12.  Given  — f-  —  =  —   to  find    x. 

D  O  ti  O 


Ans.  X  ^  \0, 

,  =  /"     to  find 

abed 


^  X  X  X  X 

13.  Given -\ "7=/    to  find     x. 


.  ahcdf 

Ans.  X  — 


U.  Given  -  -  -  ^  '^-^  -  -  12||    to  find    x. 


bed  —  acd  +  abd  —  abc 
Note. — What  is  the  numerical  value  of  x^  vrhen  a  =  1, 
u   =  2,  c  =  3,  c? r^  4,  6  :   &,  and  /=  6. 
X       8ar       a;  —  3 

Y  ""  9"  5" 

Ans.  a;  =  14. 

»c    /-.•  Sx  —  5'     4x  —  2  ,   1     ^    c  ji 

.0,  Given  a; — 1 — —  =  a;  +  1     to  find    x. 

lo  11 

Ans.  X  =  6, 

i6.  Given  x  ^  ^  +  ~  -  ^  —2x  —  4Z     to  find     x. 
4        5        6 

Ans.  X  =  60. 


/   r--         o        4a; -2       3a;  -  1     ^    .    , 
{,  Given  2a: —  =  — - —     to  find    x. 


Ans.  a;  =  3. 


1 8.  Given  3a;  +  — ^ —  =  a;  +  a     to  find    x. 

o 

.  Sa  +  li 

,r^    /^-         aa;  —  6   ,    a       5a;       ^«  —  «    x    ^  j 

19.  Given  — \-  —■  z=—- —    to  find    x, 

4  o  Z  o 

Ans.  X  = 


3a —  2^ 


KQ-JAT10N8     r)F     THE     FIRST    DKOREK.  99 

20.  Find  the  value  of  x  in  the  equation 

(^±A)_(^  _  3a  =  Mjli!  _  2.  +  «-l^' . 
a  —  6  a  -\-  h  b 

_  a'-  -f-  Sa^  -h  4a^6^  —  606^  -f  26* 

Q/'  Propositions  giving   rise  to  Equations  of  the  Firal 
Degree  involving  hut  one  unknown  quantity. 

71.  It  has  already  been  observed  (Art.  62),  that  the 
Bolution  of  a  problem  by  algebra,  consists  of  two  distinct 
parts  : 

1st.  To  make  the  statement  :  that  is,  to  express  the  con- 
ditions of  the  proposition  algebraically  ; 

2d.  To  solve  the  resulting  equation :  that  is,  to  disengage 
the  known  from  the  unknown  quantities. 

We  have  already  explained  the  manner  of  finding  the 
value  of  the  unknown  quantity,  after  the  proposition  has 
beer;  stated.  It  only  remains  to  point  out  the  best  methods 
of  stating  the  proposition  in  the  language  of  algebra. 

This  part  of  the  algebraic  solution  of  a  problem  cannot, 
like  the  second,  be  subjected  to  any  well  defined  rule. 
Sometimes  the  enunciation  of  the  proposition  furnishes  the 
equation  immediately  ;  but  sometimes  it  is  necessary  to 
discover,  from  the  enunciation,  new  conditions  from  which 
an  equation  may  be  deduced. 

71.  Into  bow  many  parts  is  the  resolution  of  a  problem  in  algebra 
divided !  What  is  the  first  step  !  What  the  second  \  Which  part  has 
already  been  explained  f  Which  part  is  now  to  be  considered  ?  Can 
this  part  be  subjected  to  exact  rules  t  Give  the  general  rule  for  stating 
a  prupoaitiua 


100  ELEMENTARY     ALGEBRA. 

In  almost  all  cases,  however/ we  are  able  to  make  the 
statement;  that  is,  to  discover  the  equation,  by  applying 
the  following 

RULE. 

Represent  the  unknown  qumitity  hy  one  of  the  final  letters 
of  the  alphabet ;  and  then  indicate  hy  means  of  the  algshrau 
siyns,  the  same  operations  on  the  known  and  unknown  '^uan 
titieSf  as  would  verify  the  value  of  the  unknown  quantity^ 
were  such  value  kjiown. 

QUESTIONS. 

1.  To  find  a  number  to  which  if  5  be  added,  the?  aum  will 
be  equal  to  9. 

Denote  the  number  by     x. 
Then  by  the  conditions 

a:  +  5  =  9. 
This  is  the  statement  of  the  proposition. 
To  find  the  value  of  ic,  we  transpose  5  to  the  second 
member,  which  gives 

a;  =  9  -  5  =  4. 

Verification. 
4  +  5  =  9. 

2.  Find  a  number  such,  that  the  sum  of  one-half,  one- 
third,  and  one-fourth  of  it,  augmented  by  45,  shall  be  equal 
to  448. 

Let  the  required  number  be  denoted  by    x. 

Then  one-half  of  it  will  be  denoted  by 


one-third  "  "  by 

une-fourtli         "  "  by 


EQUATIONS     OF     THE     FIRST     DEGREE.         101 

And  by  the  conditions, 

XXX 

2        3        4 

This  is  the  statement  of  the  proposition. 
To  find  the  value  of  ar,  subtract  45  from  both  members . 
this  gives 

XXX 
_  +  _+-=:  403. 

By  clearing  the  equation  of  denominators,  we  obtain 

6a;  +  4a;  +  3ar  =  4836, 
or  13a:  =  4836. 

IT  4836       ^^^ 

Hence,  x  =  — — -  =  372. 

lo 

Verification. 

^  +  ^  +  ^  +  45  =  186  +  124  -h  93  +  45=448 

<6  o  4 

3.  What  number  is  that  whose  third  part  exceeds  ite 
fourth  by  16? 

Let  the  required  number  be  represented  by  x.     Then, 

—  a;  =     the  third  part. 

o 

—  X  =     the  fourth  part. 
And  from  the  conditions  of  the  problem 

Tliis  is  the  statement.     To  find  the  value  of  ar,  we  clear 
the  equation  of  the  denominators,  which  gives 

4x-3a:=  192. 
and  x=  192. 


102  ELEMBNTARY     ALGEBRA. 

Verijication. 

o  4 

4.  Divide  $1000  between  A,  B  and  C,  so  that  A  shall 
hayc  172  more  than  B,  and  C  $100  more  than  A. 

Let  X  =  B's  share  of  the  $1000. 

Ther  a;  +    T2  =     A's  share, 

and  a;  4- 172  =     C's  share, 

their  sum  is         3a;  -f  244  =$1000. 

This  is  the  statement. 

By  transposing  244  we  have 

3a;  =  1000 -244  =  756 

75G 
and  X  =  -^^  =  252  =  B's  share. 

o 

Hence,  a;  -f    72  =  252  +    72  =  $324  =     A's  share. 

And  a;  +  172  =  252  +  172  =  $424  =     C's  share. 

Verijication. 
252  +  324  +  424  =  1000. 

5.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third 
part,  21  gallons  were  afterwards  di-awn,  and  the  cask  being 
then  guaged,  appeared  to  be  half  full :  how  much  did  it 
hold  1 

Suppose  the  cask  to  have  held     x    gallons. 

__  X 

Then,         —   what  leaked  away. 

o 

X 

And  •—  -f-  21  =    what  had  leaked  and  been  drawn, 

o 

X  X 

Hence,      ~  +  21  =  — -    by  the  conditions. 
This  is  tlie  statement. 


EQUATIONS     OF     THE     FIRST     DEGIIEE.         lOH 

To  find  a?,  we  have 

2x+\26  =  3ar, 
and  —    X    =  —  126, 

and  by  changing  the  signs  of  both  members,  which  does  not 
destroy  their  equality,  we  have 

X  =  126. 

Verification, 

6.  A  fish  was  caught  whose  tail  weighed  9^.,  his  head 
weighed  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighed  as  much  as  his  head  and  tail  together  ;  what  was 
the  weight  of  the  fish  1 

Let  2x  =  the  weight  of  the  body. 

Then,  9  4-^=  weight  of  the  head  ; 

and  since  the  body  "weighed  as  much  as  both  head  and  tail, 

2x  =  9  -\- 9  -\-  x, 

which  is  the  statement.     Then, 

2x  —  X  =\Q     and     x  =z  18. 

Hence  we  have, 

2x  =  SGlb.  c=  weight  of  the  body. 

9  +  X  =  21lb.  =  weight  of  the  head. 

9lb.  =  weight  of  the  tail. 

lleuco  72/6.  =  weight  of  the  fish. 


104  ELEMENTARY     ALGEBRA. 

7.  The  sum  of  two  numbers  is  67  and  their  difference 
19  :  what  are  the  two  numbers  1 

Let  X  =  the  less  number. 

Then,  a?  +  19  =  the  greater, 

and  by  the  conditions 

2a;  4-  19  =  67. 
This  is  the  statement. 

To  find  a;,  we  first  transpose  19,  which  gives 
2a:  =  67  -  19  =  48  ; 

48 
hence,        a;  =  —  =  24,    and    a;  +  19  =  43. 

Verijlcation, 
43  -h  24  =  67,    and    43  -  24  =  19. 

Another  Solution. 
Let  X  denote  the  greater  number  : 
then  a;  —  19  will  represent  the  less, 

and,  2a;  -  19  =  67,  whence  2a;  =  67  +  1^  J 

therefore,  a;  =  —  —  4rf, 

and  consequently  x—  19  =  43  —  19  =  24. 

General  Solution  of  this  Problem., 

The  sum  of  two  numbers  is   a,  their  difference   is    h. 
What  are  the  two  numbers  ? 


EQUATIONS     OF     THE     FIK8T     DEGREE.         105 

Let  X  denote  the  least  uumber. 

TheD,  X  -\-  b  will  represent  the  greater. 

Hence,         2x  -{-  b  =  a^      whence      \lx  =.  a  —  b\ 

a  —  h        a        b 
therefore,  x  =  —^  =  _  _  _  , 

a        b  a        b 

and  consequently,    ar-}-6  =  — -  —  —  +  6  =  —  -f-r- 

As  the  form  of  these  two  results  is  independent  ot  the 
values  attributed  to  the  letters  a  and  6,  it  follows  that, 

Knowing  the  sum  and  difference  of  two  numbers^  we  obtain 
the  greater  by  adding  the  half  difference  to  the  half  sum,  ant* 
the  less,  by  subtracting  the  half  difference  from  half  the  sum,* 

Thus,  if  the  given  sum  were  237,  and  the  difference  911 

.     237       99*         237-1-99      336 
the  greater  is    —  -j-  —   or    — ^—  =  —  =  168 ; 

^   ^    ,  237       99  138       ^^ 

and  the  least    -^r ir  >  or  — —  =  69. 

,6  4t  tL 


Verification, 
168  4-  69  =  237    and    168  -  69  =  99. 

8.  A  person  engaged  a  workman  for  48  days.  For  each 
day  that  he  labored  he  received  24  cents,  and  for  each  day 
that  he  was  idle,  he  paid  12  cents  for  his  board.  At  the 
end  of  the  48  days,  the  account  was  settled,  when  the  laborei 
received  504  cents.  Required  the  number  of  working  days, 
and  tJie  number  of  days  he  was  idle. 


10()  ELEMENTARY     ALGEBRA. 

If  the  two  numbers  were  known,  and  the  first  multiplied 
by  24,  and  the  second  by  12,  the  difference  of  these  pro- 
ducts would  be  504.  Let  us  indicate  these  operations  by 
means  of  algebraic  signs. 

Let  X  =  the  number  of  working  days 

48  —  a;  =  the  number  of  idle  days 
Then      24  x  a;  =  the  amount  earned 
and  12(48— re)  =  the  amount  .paid  for  board. 
Then,    24x  —  12(48  -  x)  =  504 
what  was  received,  which  is  the  statement. 
Then  to  find  x  we  first  multiply  by  12,  which  gives 

24j;  — 576  +  12^  =  504. 
or,  30^  =  504  +  57G  =  1080, 

and  X  =  — — -  =  30  the  number  of  working  days : 

whence,  48  —  30  =  18      the  number  of  idle  days; 

Verification, 

Tliirty  days'  labor,  at  24  cents 
a  day,  amounts  to 30  X  24  =  720  cents. 

And  18  day's  board,  at  12  cents 
a  day,  amounts  to 18  X  12  =  216  cents. 

The  difference  is  the  amount  received  504  cents. 

General  Solution, 

This   problem   may  be   made  general,  by  denoting  the 
whole  number  of  working  and  idle  days  by  n. 
The  amount  received  for  each  day's  work  by  a, 
Tlie  amount  paid  for  board,  for  each  idle  day,  by  b. 


EgUATIONS     OF     THE     FIRST     DKORKR.         107 

And  the  balance  due  the  laborer,  or  the  result  of  the 
ftcccunt,  by  c. 

As  before,  let  the  number  of  working  days  be  denoted 
by  X. 

The  number  of  idle  days  will  then  be  expressed  by  »— as. 

Hence,  what  is  earned  will  be  expressed  by  ax. 

And  the  sum  to  be  deducted,  on  account  of  board,  by 
b{n  -  x). 

The  statement  of  the  problem,  therefore,  is 


ax  —  b{n  —  x)  =  c. 

To  find  ar, 

we 

first 

,  multiply  by  6,  which  gives 
ax  —  bn  -{-  bx  =  c. 

or, 

{a-^b)x  =  c  +  bn, 

whence, 

c  +  bn 
X  =  — --  -  -  =     number  of 
a  -{-  b 

-  .  c  -{■  bn     an -\- bn  —  c  —  bn 

and  consequently,  n  —  x  =  n r-  = — r , 

^  a  -^  b  a-f-o 

IT,  n  —  X  —  — — -r-  =     number  of  idle  days, 

a  -f-  6  "^ 


Let  us  now  suppose  n  =48,  a  =  24,  6  =  12,  and  c  •=  504. 
These  numbers  will  give  for  x  the  same  value  as  before 
found. 

9.  A  person  dying  leases  half  of  his  property  to  his  wife, 
one-sixth  to  each  of  two  da\ighters,  one-twelfth  to  a  servant, 
and  the  remaining  $000  to  the  poor :  what  was  the  amount 
of  his  property  1 


l08  ELEMENTARY     ALGEBRA. 

Denote  the  amount  of  the  property  by  x. 

X 

Then         —  =         what  he  left  to  his  wife, 


what  he  left  to  one  daughter, 


and  —  =  —     what  he  left  to  both  daughters , 

O  o 

X 

also  —  =1         what  he  left  to  his  servant. 

and  $600  what  he  left  to  the  poor. 

Then^  by  the  conditions 

XXX 

—  4-  -^  +  7-^  +  600  =  X  the  amount  of  the  property, 

which  gives     x  =  17200. 

10.  A  and  B  play  together  at  cards.  A  sits  down  with 
$84  and  B  with  $48.  Each  loses  and  wins  in  turn,  when 
it  appears  that  A  has  five  times  as  much  as  B.  How  mujh 
did  A  win  1 

Let  X  represent  what  A  won. 
Then,  A     rose  with     $84  +  x     dollars, 

and  B     rose  with     $48  —  x     dollars. 

But  by  the  conditions,  we  have 

84  -f  a:  =  5(48  -  x), 
hence,  84  -f  a;  =  240  -  5a: ; 

and,  '  6a:  =  156, 

consequently,  x  =  $26     what  A  won. 

Verijlcation, 

84  +  26  =  110  ;     48  -  26  =  22 ; 
110  =  5(22)  =  110. 


EQUATIONS     OF     THE     FIRST     DEGREE 


109 


11.  A  can  do  a  piece  of  work  alone  in  10  days,  B  in  13 
days ;  in  what  time  can  ^hey  do  it,  if  they  work  together] 
Denote  the  time  by  ar,  and  the  work  by  1.     Then,  in 

1  day,  A  can  do  —  of  the  work,  and 


B  can  do  —  of  the  work ;  and  In 

lo 
X 

X  days,  A  can  do  —  of  the  work,  and 


B  can  do  — -  of  the  work : 
lo 

hence,  by  the  conditions 

-^  -f  :^  =  1,     which  gives     13ar  -f-  lOar  =  130 
10       lo 

hence,         23jr  =  130, 


a;  =  — -  =  5if  days. 


23 


12.  A  fox,  pursued  by  a  greyhound,  has  a  start  of  60 
leaps.  He  makes  9  leaps  while  the  greyhound  makes  but 
0 ;  but,  3  leaps  of  the  greyhound  are  equivalent  to  7  of  the 
fox.  How  many  leaps  must  the  greyhound  make  to  over- 
take the  fox  1 

From  the  enunciation,  it  is  evident  that  the  distance  to 
be  passed  over  by  the  greyhound  is  composed  of  the  GO 
leaps  which  the  fox  is  in  advance,  plus  the  distance  that  the 
fox  passes  over  from  the  moment  when  the  greyhound  starts 
in  pursuit  of  him.  Hence,  if  we  can  find  the  expressions 
for  these  two  distances,  it  will  be  easy  to  state  the  problem. 

Let  X  =  the  number  of  leaps  made  by  the  greyhound 
before  he  overtakes  the  fox. 

Now,  since  the  fox   makes  9  leaps  while  the  greyhound 

9  3 

makes  but  6,  t^**^  f<  x   will  make    —    or    --    leaps    N^hilo 


110  ELEMENTARY     ALGKllKA. 

the  greyhound  makes  1 ;  and,  therefore,  while  the  gre}hr-uiif' 

Sx 
makes  x  leaps,  the  fox  will  make    —     leaps. 

Hence,  the  distance  which  the  greyhound  must  pass  oYei 

Sx 

will  be  expressed  by  60  +  —    leaps  of  the  fox. 

It  might  be  supposed,  that  the  equation  might  be  obtaineo 

3 
by  merely  placing  x  equal  to  CO  +  —  a* ;    but  in  doing  so,  a 

manifest  error  would  be  committed ;  for  the  leaps  of  the  grey 

hound  are  greater  than  those  of  the  fox,  and  we  should  ther 

equate  numbers  of  different  denominations ;  that  is,  num 

bers  having  different  units.     Hence,  it  is  necessary  to  ex 

press  the  leaps  of  the  fox  in  terms  of  those  of  the  grey 

hound,  or  reciprocally.     Now,  according  to  the  enunciation 

3  leaps  of  the  greyhound  are  equivalent  to  7  leaps  of  thi 

7 
fox  ;  then,  1  leap  of  the  greyhound  is  equivalent  to  —  leap^. 

o 

of  the  fox  ;  and  consequently,  x  leaps  of  the  greyhound  ar^ 

7x 
equivalent  to  — -  of  the  fox's  leaps. 

o 

Hence,  we  have  the  equation 

Making  the  denominators  disappear 

14a;  =  3G0  -f  9a;, 
whence,  6x  =  3G0     and    x  =  72  : 

Therefore,  the  greyhound  will  make*  72  leaps  before  over 
taking  the  fox,  and  during  this  time,  the  fox  will  make 

72  X  —    or    lOS  leaps. 


liQUATIONb     OF     THii    vHS^JbOlii  ^  K  £  .         1 1 J 


l'"rfJicatiofi. 

The  72  leaps  of  uie  gixv hound  arc  equivaloiit  to 

72  :>^  7 

— ^  =  168  leaps  of  the  foi, 
3 

And  60  4  108  =  168, 

the  leaps  which  the  fox  made  from  the  beginning, 

13.  A  father  leaves  his  propertj-,  amounting  to  12520,  to 
four  sons,  A,  B,  C,  and  D.  C  is  to  have  $300,  B  as  much 
as  C  and  D  together,  and  A  twice  as  much  as  B  less  |1000' 
how  much  do  A,  B,  and  D  receive  1  ^ 

Ans,  A,  $760,  B,  $880,  D,  $520 

14.  An  estate  of  $7500  is  to  be  divided  between  a  widow 
two  sons,  and  three  daughters,  so  that  each  son  shall  reoeivf 
twice  as  much  as  each  daughtc^r,  and  the  widow  herself  $500 
more  than  all  the  children :  what  was  her  share,  and  what 
the  share  of  each  child? 

(  Widow's  share,   $4000. 

Ans.  }  Each  son's,  $1000. 

(  Each  daughter's,  $  500. 

15.  A  company  of  180  persons  consists  of  men,  women, 
and  children.  The  men  are  8  more  in  number  than  the 
women,  and  the  children  20  more  than  the  men  and  women 
together :  how  many  of  each  sort  in  the  company  1 

Ans.  44  men,  30  women,  100  children. 

16.  A  father  divides  $2000  among  five  sons,  so  that  each 
elder  should  receive  $40  more  than  his  next  younger  bro- 
ther :  what  is  the  share  of  the  youngest  1  Ans.  $320 

17.  A  purse  of  $2850  is  to  be  divided  among  three  per 
SOILS,  A,  D,  and  0.     A's  share  is  to  bo  to  B's  a:^  0  to  11 


il2  ELEMENTARY     ALGEBRA. 

and  C  is  to  have  $300  more  than  A  and  B  together :  what 
is  each  one's  share  1  Ans.  A's  $450,  B's  $825,  C's  $1575. 
18.  Two  pedestrians  start  from  the  same  point ;  the  first 
steps  twice  as  far  as  the  second,  but  the  second  malces  5 
steps  while  the  first  makes  but  one.  At  the  end  of  a  cer- 
tain  time  they  are  300  feet  apart.  Now,  allowing  each  of 
the  longer  paces  to  be  3  feet,  how  far  will  each  have  trav- 
elled 1  .  Ans,  1st,  200  feet;  2d,  500. 

>J  19.  Two  carpenters,  24  journeymen,  and  8  apprentices, 
received  at  the  end  of  a  certain  time  $144.  The  carpen- 
ters received  $1  per  day,  each  journeyman,  half  a  dollar, 
and  each  apprentice  25  cents :  how  many  days  were  they 
employed  1  Ans.  9  days. 

^    I  20.  A  capitalist  receives  a  yearly  income  of  $2940  :  four- 

T^ths  of  his  money  bears  an  interest  of  4  per  cent,  and  the 
remainder  of  5  per  cent :  how  much  has  he  at  interest  1 

Ans.  70000. 

21.  A  cistern  containing  60  gallons  of  water  has  three 
unequal  cocks  for  discharging  it ;  the  largest  will  empty  it 
in  one  hour,  the  second  in  two  hours,  and  the  third  in  three  ; 
in  what  time  wdll  the  cistern  be  emptied  if  they  all  run 
together  1  Ans.  32  j\  mi/i. 

22.  hi  a  certain  orchard,  one-half  are  apple  trees,  one- 
fourth  peach  trees,  one-sixth  plum  trees  ;  there  are  also,  120 
cherry  trees,  and  80  pear  trees :  how  many  trees  in  the 
orchard  1  Ans.  2400. 

23.  A  farmer  being  asked  how  many  sheep  he  had,  an- 
swered,  that  he  had  them  in  five  fields  ;  in  the  1st  he  had  i, 
in  the  2d,  ^,  in  the  3d,  ^,  and  in  the  4th,  ^^^  ^^^  ^^  ^h®  ^t^> 
450  :   how  many  had  he  ?  Ans.  1200. 

24.  My  horse  and  saddle  together  are  worth  $132,  and 
the  horse  is  worth  ten  times  as  much  as  the  saddle  :  what 
h  the  value  of  the  horfic?  Ans,  120. 


KliUATIO.  THE     FIRST     DKOKKE.         113 

25.  The  rent  of  an  estate  is  this  year  8  per  cent  greater 
than  it  was  last.  This  year  it  is  $1890  :  what  was  it  last 
ye&r'i  ^«s.  $1750. 

2G.  What  number  is  that  from  which,  if  5  be  subtracted, 
{  of  the  remainder  will  be  40  1  Ahm.  05. 

27.  A  post  is  J  m  the  mud,  J  in  the  water,  and  10  feot 
»bovo  the  water:  what  is  the  whole  length  of  the  post? 

A718.  24  feet. 

28.  After  paying  J  and  J  of  uiy  money,  1  had  60  guineas 
left  in  iMV  r>nr^p  •   how  many  guineas  were  in  it  at  first? 

Ans.  120. 
\^20.  A  person  was  desirous  of  giving  3  pence  apiece  to 

'^mc  beggars,  but  found  he  had  not  money  enough  in  his 
pocket  by  8  pence;  he  therefore  gave  them  each  2  pence 
4nd  had  3  penc^  remaining  :  required  the  number  of  beg- 
gars. Atis.  11. 

»  /  30.  A  person,  in  play,  lost  \  of  his  money,  and  then  won 
3  shillings ;  after  which  he  lost  J  of  what  he  then  had  ;  and 
this  done,  found  that  ho  hud  hut  12  shilliii'>s  lemaining; 
what  had  he  at  first  1  A71S.  20t, 

31.  Two  persons,  A  and  15,  lay  out  e4ual  .suins  of  money 
in  trade ;  A  gains  $126,  and  B  loses  $87,  and  A's  money 
is  now  double  of  B's  :  what  did  each  lay  out  1  Ans.  $300. 
'  32.  A  person  goes  to  a  tavern  with  a  certain  sura  of 
money  in  his  pocket,  where  he  spends  2  shillings :  he  then 
borrows  as  much  money  as  he  had  left,  and  going  to  another 
tavern,  he  there  spends  2  shillings  also;  then  borrowing 
again  as  much  money  as  wAs  left,  he  went  to  a  third  tavern, 
where  likewise  he  spent  2  shillings  and  borrowed  as  much 
as  he  had  left ;  and  again  spending  2  shillings  at  a  fourth 
'\rrn,  he  then  had  nothing  remaining.     What  had  he  at 

AiiM.  iis.  ifd. 


114  MLEMENTARY     ALGEBRA. 

Of  Equations  of  the  First  Degree  involving  two  Oi  nwrt 
unknown  quantities. 

72.  Several  of  the  problems  already  discussed  nave 
apparently  involved  more  than  one  unknown  quantity  ;  yet 
we  have  been  able  to  solve  them  all  by  the  aid  of  a  single 
unknown  symbol.  In  these  cases,  the  required  parts  of  the 
problem  have  been  so  connected  that  we  have  been  able  to 
express  the  relations  between  them  by  means  of  a  single 
equation.  We  come  now  to  those  problems,  in  the  solution 
of  which,  we  employ  more  than  one  unknown  quantity. 

Let  us  first  examine  some  of  those  problems  which  we 
have  already  solved  by  the  aid  of  but  a  single  unknown 
symbol. 

1.  Given  the  sum  of  two  numbers  equal  to  36,  and  their 
difference  equal  to  12,  to  find  the  numbers. 

Let  X  =  the  greater,  and  y  =  the  less  number. 
Then,  from  the  1st  condition    .     .     .     .     x  -{-  y  =z  36, 
and  from  the  second, x  —  y  =  12. 

By  adding  (Art.  65,  Ax.  1),     ....         2  a;  =  48. 
By  subtracting  (Art.  65,  Ax.  2),  .     .     .  2y  =  24. 

Each  of  these  equations  contains  but  one  unknown  guan 
tity. 

48 
From  the  first,  we  obtain     ,     .     .  .  ar  =  —  =  24. 

24 

And  from  the  second, y  =  —-  =z  V>L 

Z 

Verification. 

ar  -f  y  =  30     gives     24  +  12  =  36, 
x-y^\2        "        V4~12=V^. 


BQUATIONS     OF     THE     FIRST      DEUKEE.  115 

General  Solution. 
Let  X  =  the  greikter,  and  y  the  less  number. 

Then  by  the  conditions x  •\-  y  =  a, 

and X  —  y  z=b. 

By  adding,  (Art.  65,  Ax.  1), 'Hx  =  a  +  b. 

By  subtracting,  (Art.  65,  Ax.  2),   .     .     .     ,  2y  =  a  —  b. 
Each  of  these  equations  contains  but  one  unknown  quantity. 

a  -f  6 
From  the  first,  we  obtain x  =  — ^^* 

a  —  b 
And  from  the  second, y  =  — ^r — . 

Verification. 

a-i-  b       a  —  b       2a  .    a -\-  b       a  —  b       2b       , 

4- =  —  ^  a  :  and =  —  =  o, 

2^2  2  '  2  2  2 

For  a  second  example,  let  us  also  take  a  problem  that 
has  been  already  solved. 

2.  A  person  engaged  a  workman  for  48  days.  For  each 
day  that  he  labored  he  was  to  receive  24  cents,  and  for  each 
day  that  he  was  idle  he  was  to  pay  12  cents  for  his  board. 
At  the  end  of  the  48  days  the  account  was  settled,  when  the 
laborer  received  504  cents.  Required  the  number  of  work 
ing  days,  and  the  number  of  days  he  was  idle. 
I^t  X  =     the  number  of  working  days, 

y  =     the  number  of  idle  days. 
Then  242r  =     what  he  earned, 

and  12y  =     what  he  paid  for  his  board. 

Then,  by  the  conditions  of  the  question,  we  have 

x-^y      =48, 
and  24a;  -  12y  =  504. 

lliiis  13  tlio  statement  of  tlie  problem. 


116  KLEMENTAKY     ALGEBRA. 

It  has  already  been  shown  (Art.  65,  Ax.  3),  that  the  two 
members  of  an  equation  can  be  multiplied  by  the  same 
number,  without  destroying  the  equality.  Let,  then,  the 
first  equation  be  multiplied  by  24,  the  co-efficient  of  a:  in 
the  second :  we  shall  then  have 

24a; +  24y  =  1152, 
24a;  — 12y=    504, 

And  by  subtracting,  36y  =    648, 

and  y  =  ^  =  is. 

Substituting  this  value  of  y  in  the  equation 

24a;  —  12y  =  504,     we  have    24a;  —  216  =.-  504, 
vhich  gives 

720 


24a; 

=  504  +  216^=  720,     and    x 

~  24   - 

30. 

Verification, 

a;  + 

y  = 

48 

gives 

30  +  18 

=    ^^ 

24a;- 

12y  = 

504 

gives     24  X  30  - 
Elimination. 

- 12  X  18 

=  504. 

73.  The  process  of  combining  two  or  more  equations,  in- 
volving two  or  more  unknown  quantities,  and  deducing  there- 
from a  single  equation  involving  but  one,  is  called  elimina^ 
lion. 


73.  What  is  elimination  ?  How  many  methods  of  elimination  are 
_;ljere  ?  Give  the  rule  for  elimination  by  addition  and  subtraction.  What 
's  the  fhst  otep  ?     Wliat  the  second  ?     What  the  third  f 


SQUATIONb      OF     THE      FI}CSr      UKOilKE  117 

There  are  three  principal  methods  of  elimination : 

1st.  By  addition  and  subtraction. 

2d.    By  substitution. 

3d.    By  comparison. 
We  will  consider  these  methods  separately. 

Elimination  hy  Addition  and  Subtraction, 

1.  Take  the  twf>  equations 

3ar  —  2y  =  7 
Sir  -h  2y  =  48. 

If  we  add  these  two  equations,  member  to  member,  9fe 
obtain 

liar  =  55: 

which  gives  by  dividing  by  1 1 

a;  =  5: 

and  substituting  this  value  in  either  of  the  given  equations, 
we  find 

y  =  4. 

2.  Again,  take  the  equations 

8jr  -h  2y  =  48 
3x  +  2y  =  23. 

If  we  subtract  the  2d  equation  from  the  first,  we  obtain   , 

5jr  =  25, 

which  gives,  by  dividing  by  5, 

and  by  substituting  this  value,  we  fmd 


118  ELEMENTARY     ALGEBRA. 

3.  Take  the  two  equations 

5;c  +  7y  =  43. 
lla;  +  9y  =  G9. 
If,  in  these  equations,  one  of  the  unknown  quantities  was 
affected  with  the  same  co-efficient,  we  might,  by  a  simple 
subtraction,  form  a  new  equation   which  would  contain  but 
one  unknown  quantity. 

Now,  if  both  members  of  the  first  equation  be  multiplied 
by  9,  the  co- efficient  of  y  in  the  second,  and  the  two  mem- 
bers of  the  second  by  7,  the  co-efficient  of  y  in  the  first,  we 
will  obtain 

45a;  +  63y  =  387, 
77«  +  63?/  =  483. 

Subtracting,  then,  the  first  of  these  equations  from  the 
second,  there  results 

32ar  =  96,     whence     a;  =  3. 

Again,  if  we  multiply  both  members  of  the  first  equation 
by  11,  the  co-efficient  of  x  in  the  second,  and  both  members 
of  the  second  by  5,  the  co-efficient  of  x  in  the  first,  we  shall 
form  the  two  equations 

55a:  +  77y  =  473, 
55a;  +  45y  =  345. 

Subtracting,  then,  the  second  of  these  two  equations  from 
the  first,  there  results 

32y  =  128,     whence     y  =  4. 
Therefore  a;  =  3  and  y  =  4,  are  the  values  of  x  and  y. 

Verification, 

5a;  4  "^y  =  43    gives      5x3  +  7x4=  15 +  28  =  43; 
U2;  +  05  =  69      "        11x3  +  9x4  =  33  +  36  =  69. 


EQUATIONS     OF     THE     FIRST     DEOIIEE.         ll*J 

The  method  of  elimination  just  explained,  is  called  the 
method  by  addition  and  subtraction. 

To  eliminate  by  this  method  we  have  the  following 

RULE. 

I.  See  which  oj  the  unknown  quantities  you  will  eliminate, 

II.  Make  the  co-ejfficienls  of  this  unknown  quantity  equal  in 
the  two  equations^  either  by  multiplication  or  division. 

III.  If  the  siyns  of  the  like  terms  are  the  same  in  both 
equations^  subtract  one  equation  from  the  other  ;  but  if  the 
tigns  are  unlike^  add  them. 

EXAMPLES. 

4.  Find  the  values  of  x  and  y  in  the  equations 

3a?  —  y  =  3, 

y  -f  2ar  =  7. 

Ans.  a;  =  2,  y  E=  8. 

5.  Fmd  the  values  of  x  and  y  in  the  equations 

^x-ly=  -  22, 
5x-\-2y  =  37. 

Ans.  a;  =  5,  y  =  6. 

0.  Find  the  values  of  x  and  y  in  the  equations 

2ar  +  6y  =  42, 
8a;  — 6y=    3. 

Ans.  a;  =  4J,  y  =  5}. 

7.  Find  the  values  of  x  and  y  in  the  equations 

8a:-9y=  1, 
Gar  —  3y  =  4a;. 

Ans.  a;  =  J,  y  =  J. 


-a;  +  -y  =  6, 


-x  +  -7/z=Gl. 


120  ELKMKNTARt      ALGEBRA. 

8.  Find  the  values  of  x  and  y  in  the  equations 

14a;  -  15y  =  12, 
7x4-    8y  =  37. 

Ans.  X  =  S^  y  i=.*i 

9.  Find  the  values  of  x  and  y  in  the  equations  * 

1  2. 

3 

i.  +  l 

Ans.  a;  =  0,  y  =  9. 

10.  Find  the  values  of  x  and  y  in  the  equations 

a:  -  y  =  -  2. 

Ans,  X  =  14:,  y  z=z  16. 

11.  Says  A  to  B,  you  give  me  $40  of  your  money,  and 
I  shall  then  have  5  times  as  much  as  you  will  have  left. 
Now  they  both  had  $120  :  how  much  had  each  ? 

Ans.  Each  had  $60. 

12.  A  father  says  to  his  son,  "  twenty  years  ago,  my  age 
was  four  times  yours ;  now  it  is  just  double :"  what  were 
their  ages  1  A  s  \  I^^^her's,  60  years. 

*  (  Son's,        30  years. 

13.  A  father  divides  his  property  between  his  two  sons.. 
At  the  end  of  the  first  year  the  elder  had  spent  one-quarter 
of  his,  and  the  younger  had  made  $1000,  and  their  property 
was  then  equal.  After  this  the  elder  spends  $500  and  the 
younger  makes  $2000,  when  it  appears  the  younger  has  just 
double  the  elder  :  what  had  each  from  the  father  1 


.       (  Elder,       $4000. 
'^^'  I  Younger,  $2000. 


EQUATIONS     OF     TUB     FIRST     DEORKE.         121 

14.  If  John  give  Charles  15  apples,  they  will  havi.  the 
same  number;  but  if  Charles  give  15  to  John,  John  will 
have  15  times  as  many,  wanting  10,  as  Charles  will  have 
left.     IIow  many  had  each  ?  .        j  John       50. 

1  Charles  20. 

15.  Two  clerks,  A  and  B,  have  salaries  which  are  together 
equal  to  $900.  A  spends  ^  per  year  of  what  he  receives, 
and  B  adds  as  much  to  his  as  A  spends.  At  the  end  of  the 
year  they  have  equal  sums :  what  was  the  salary  of  each  1 

.        j  A's  =  500. 
(  B's  =  400 


Elimination  by  SuhsiihUion, 

74.  Let  us  again  take  the  equations 

5x  +  7y  =  43, 
11x4-0^  =  69. 

Find  the  value  of  x  in  the  first  equation,  which  gives 
43 -Ty 


X  = 


5 


SubstituA>e  this  value  of  x  in  the  second  equation,  and  we 
have 

or,  473  -  77y  +  45y  =  345, 
or,  ~  32y  =  -  128. 

Ilenoe,  y  =  4, 

A  43  -  28      ., 


122  ELEMENTARY     ALQEliRA. 

This  method  is  called  the  method  by  substitution:  we 
have  fcr  the  process  the  following 

RULE. 

Mnd  the  value  of  one  of  the  unknown  quantities  from 
either  of  the  equations  ;  then  substitute  this  value  in  the  other 
equation :  there  will  thus  arise  a  new  equation  with  but  one 
unknoum  quantity. 

Kemark. — This  method  of  elimination  is  used  to  great 
advantage  when  the  co-efficient  of  either  of  the  unknown 
quantities  is  unity. 

EXAMPLES. 

1.  Find,  by  the  last  method,  the  values  of  x  and  y  in  the 
equations 

Sx  —  y  =  1     and     3y  —  2a;  =  4. 

Ans,  a:  =  1,  ?/  =  2, 

2.  Find  the  values  of  x  and  y  in  the  equations 

5y  —  4x=  —22    and    Sy  +  4x  z=  38. 

Ans.  X  =  Sj  y  z=2, 

\  Find  the  values  of  x  and  y  in  the  equations 
a;  4-  8y  =  18     and    y  -  Sx  =  -  29. 

Ans.  a;  =  10,  3^  =  1. 

i.  Find  the  values  of  x  and  y  in  the  equations 
5a;  -  y  =  13     and     8x-{-~-y  =  29. 

Ans.  X  =  S^,y  —  4J. 

V  Give  the  rule  for  elimination  by  substitution  ?  When  is  it  desira- 
ble to  use  this  method  ? 


IQUATIONU     or     THK     FIUBT     DEUKEK.         123 

5.  Find  the  values  of  x  aiid  y,  from  the  equations 

10«  -  -|-  =  09     and     lOy  -  ^=  49. 

Ans.  X  =  7j  y  =  S, 

6.  Find  the  values  of  x  and  y,  from  the  equations 

^n«.  a?  =  8,  y  =  10. 

7.  Find  the  values  of  x  and  y,  from  the  equations 

|.-|  +  5=2,     .  +  f  =  17i. 

^rw.  a;  =  15,  y  =  14. 

8.  Find  the  values  of  x  and  y,  from  the  equations 

|^|^3  =  6i,    ana    f-l.!. 

^n5.  ar  =  3J,  y  r=  4. 

9.  Find  the  values  of  x  and  y,  from  the  equations 

|_±  +  e  =  5.    and    ^-1^  =  0. 

Ans.  X  =  12,  y  =  10, 

10.  Find  the  values  of  x  and  y,  from  the  equations 

y-5^-l  =  -9,     and     5x-5^  =  29. 

u4n«.  ar  =  6,  y  =  7. 

11.  Two  misers,  A  and  B,  sit  down  to  count  over  their 
money.  They  both  have  $20000,  and  B  has  three  times  as 
much  as  A  :  how  much  has  each  1 


.       (  A  .  .  15000. 
***•(«..  $15000. 


124  ELEMENTARY     ALGEBRA. 

12.  A  person  has  two  purses.  If  he  puts  $7  into  the  first 
purse,  it  is  worth  three  times  as  much  as  the  second :  but  ii 
he  puts  $7  into  the  second,  it  becomes  worth  five  times  as 
much  as  the  first :  what  is  the  value  of  each  purse  ? 

Ans.  1st,  $2  :  2d,  |3. 

13.  Two  numbers  have  the  following  properties :  if  the 
first  be  multiplied  by  6,  the  product  will  be  equal  to  the 
second  multiplied  by  5 ;  and  one  subtracted  from  the  first 
leaves  the  same  remainder  as'2  subtracted  from  the  second  : 
what  are  the  numbers  ?  Ans,  5  and  6. 

14.  Pind  two  numbers  with  the  following  properties :  the 
first  increased  by  2  to  be  3J  times  as  great  as  the  second : 
and  the  second  increased  by  4  gives  a  number  equal  to  half 
the  first :  what  are  the  numbers  1  Ans.  24  and  8. 

15.  A  father  says  to  his  son,  "  twelve  years  ago,  I  was 
twice  as  old  as  you  are  now :  four  times  your  age  at  that 
time,  plus  twelve  years,  will  express  my  age  twelve  years 
hence :"  what  were  their  ages  *?       j         j  Father,  72  years, 

^^^-    I  Son,        30      " 

JEJlimination  hy  Comparison, 

76.  Take  the  same  equations 

5:r  +  7y  =  43, 
Ux  +  ^y  =  69. 

Finding  the  value  of  x  from  the  first  equation,  we  have 

43 -7y 

X  =  -' 

5 

and  finding  the  value  of  x  from  the  second,  we  obtain 

__  69 -9y 

^■"       11      * 


EQUATIONfl     OF     THE     FIRST     DEGREE.         125 

Let  these  two  values  of  x  be  placed  equal  to  each  other, 
and  we  have 

43  —  7y  _  69  -  9y 
5        ~       11 

Or,  473  -  77y  =  345  -  45y  ; 

or,  -  32y  =  —  128. 

Hence,  y  =  4. 

69-36      ^ 
And,  X  =  — -- —  =  3. 

This  method  of  elimination  is  called  the  method  by  com 
parison,  for  which  we  have  the  following 

RULE. 

I.  Find  the  value  of  the  s^amp.  unknown  quantity  from  each 
equation. 

II.  Place  these  values  equal  to  each  other  ;  and  a  new  equa- 
tion will  arise  involving  but  one  unknown  quantity. 

EXAMPLES. 

1.  Find,  by  the  last  rule,  the  values  of  x  and  y,  from  the 
equations 

3:p  +  -|  +  6  =  42     and     y-^  =  14i. 

Ans,  ar  =  11,  y  =  15. 


76    Give  the  rule  for  elimination  by  coropanBon  t     What  is  tho  fixvf 
»tep       WLnt  the  frcoml ! 


126  ELEMENTARY     ALGEBRA. 

2.  Find  the  values  of  x  and  y,  from  the  equations 

f -7  +  5  =  6    and    |-  +  4=Jl  +  6.' 

Ans,  ic  =  28,  y  =  20. 

3.  Find  the  values  of  x  and  y,  from  the  equations 

V        a;       22 
^--  +  -  =  1     and    3,-x  =  6. 

Ans.  a;  =  9,  y  =  5. 

4.  Find  the  values  of  x  and  y,  from  the  equations 

y-3=ia;4-5    and    ^-±l=,y^^. 

Ans.  a:  =  2,  y  =  9. 
6.  Find  the  values  of  x  and  y,  from  the  equations 

L^  +  |  =  ,_.    ,„,    l  +  f  ..-13. 

u4n5.  a;  =  16,  y  =  7. 

6.  Find  the  values  of  x  and  y,  from  the  equations 

y  -^  X   ^   y  —  x  2y 

^^4-^^  =  a;--|,    and    a:  +  y  =  16. 

Ans.  a;  =  10,  y  =  6. 

7.  Find  the  values  of  x  and  y,  from  the  equations 

-47is.  ar  =  1,  y  =  3 


8.  Find  the  values  of  a^  and  y,  from  the  equations 

-4  a; 

-  =  2^-5* 
-4n5.  a;  =  ]0,  y  _  13. 


2yH-3ar  =  y4  43,    y-^-^  =  y_£ 


KQUATIONS     OF     THE     FIRST     DEGRKK.         127 

9.  Find  the  values  of  x  and  y,  from  the  equations 

^y^^LlLl-x-\-  18,  and  27-y  =  ar-f  y-f  4. 

Ana.  X  z=9j  y  =3'7. 
10.  Find  the  values  of  x  and  y,  from  the  equatioLs 

Ans,  a;  =  10,  y  =  20. 


76.  Having  explained  the  principal  methods  of  eliminar 
tion,  we  shall  add  a  few  examples  which  may  bo  solved  by 
any  one  of  them  ;  and  often  indeed,  it  may  be  advantageous 
to  employ  them  all  even  in  the  same  problem. 


GENERAL    EXAMPLES. 


V 


1.  Given  22?  +  3y  =  16,  and  3x  —  2y  =  11,  to  find  the 
values  of  x  and  y.  Ans.  a;  =  5,  y  =  2. 

^    ^.        2x      Sy       9        ^    Sx      2y       61        .      ^   :. 
2.G.vcn-  +  J^  =  -and-  +  J!  =  ^,     to    find 

the  values  of  x  and  y.  Ans.  or  =  — ?    y  =  — . 

3.  Given  -|-  +  7y  =  99,  and  "I-  -f-  7a;  =  51,    to  find  the 
values  of  x  and  y.  Atis.  x  =  1,  y  =^  14. 

4.  Given 

_-.  12  =  ^  +  8,  and  -^  +  --8  =  -^— +27, 
U    TjuI  tlio  values  of  x  and  y  ^n^.  x  =  60,  y  —  40. 


12S  JBLKMENTART     ALGEBRA, 


QUESTIONS, 


1.  What  fraction  is  that,  to  the  numerator  of  which  if  J 
be  added,  the  value  will  be  — ,    but  if  one  be  added  to  its 

o 

denominator,  the  value  will  be  —  * 

X 

Let  the  fraction  be  represented  by   — 

Then  by  the  conditions 

rc+l         1  ^  a;  1 

Whence,  3a;  -f-  3  =  y,     and,     4a;  =  y  +  1. 

Therefore,  by  subtracting, 

a;  —  3  =  1     and    a;  =  4. 
Hence,  12  -{-  3  =  y ; 

therefore,  y  =  15. 

2.  A  market-woman  bought  a  certain  number  of  eggs  at 
2  for  a  penny,  and  as  many  others,  at  3  for  a  penny ;  and 
having  sold  them  altogether,  at  the  rate  of  5  for  2cf,  found 
that  she  had  lost  4c^  :  how  many  of  both  kinds  did  she  buy  1 

Let  2x  =     the  whole  number  of  eggs. 

Then  x  =     the  number  of  eggs  of  each  sort. 

Then  will         tt  ^  =     the  cost  of  the  first  sort, 

and  "q"  ^  =     ^^  ^^st  ^^  ^^  second  sort. 

o 


But  by  the  conditions  of  the  question    5  :  2a;  :  :  2 
the  amount  for  whicli  the  eggs  were  sold. 


If 
5' 


KQUATI0N8     OF     THE     FIRST     DEGREE.         120 

Hence,  by  the  conditions 

Therefore,  15a:  -f  IOj?  —  24ar  =  120 

or  x  =  120 ; 

the  number  of  eggs  of  each  sort. 

3.  A  person  possessed  a  capital  of  30,000  dollars,  for 
which  he  drew  a  certain  interest ;  but  he  owed  the  sum  of 
20,000  dollars,  for  which  he  paid  a  certain  interest.  The 
interest  that  he  received  exceeded  that  which  he  paid  by 
800  dollars.  Another  person  possessed  35,000  dollars,  for 
which  he  received  interest  at  the  second  of  the  above  rates ; 
but  he  owed  24,000  dollars,  for  which  he  paid  interest  at 
the  first  of  the  above  rates.  The  interest  that  he  received 
exceeded  that  which  he  paid  by  310  dollars.  Required  the 
two  rates  of  interest. 

Let  x  and  y  denote  the  two  rates  of  interest ;  that  is,  the 
interest  of  $100  for  the  given  time. 

To  obtain  the  interest  of  $30,000  at  the  first  rate,  denoted 
by  X,  we  form  the  proportion 

100  :  X  ::  30,000    :    ^-^^     or     300:r. 

And  for  the  interest  $20,000,  the  rate  being  y, 

100  :  y  : :  20,000    :    ?^^     or     200y. 

But  by  the  conditions,  the  difference  between  these  rwo 
amounts  is  equal  to  800  dollars. 

We  have,  then,  for  the  first  equation  of  the  problem 

300;r  -  200y  =  800. 
0* 


ISO  KLEMEliTARr     ALGEBRA. 

V 

By  expressing  algebraically,  the  second  condition  of  the 
pioblem,  we  obtain  the  other  equation, 

350y  -  240a;  =  310. 

Both  members  of  the  first  equation  being  divisible  by 
100,  and  those  of  the  second  by  10,  we  have 

Sx  —  2i/=  8,         35y  —  24a;  =  81. 

To  eliminate  a?,  multiply  the  first  equation  by  8,  and  theo 
add  the  result  to  the  second  :  there  results 

I9y  =  95,     whence    y  =  5. 

Substituting  for  y,  in  the  first  equation,  this  value,  and 
that  equation  becomes 

Sa;  —  10  =  8,     whence     x  =  Q. 

Therefore,  the  first  rate  is  6  per  cent,  and  the  second  5. 

Verification, 

$30,000,    placed  at  6  per  cent,  gives     300  X  6  =  $1800. 
$20,000,        "  5         "  "        200  X  5  =  $1000. 

And  we  have  1800  —  1000  =  800. 

The  second  condition  can  be  verified  in  the  same  manner. 

4.  What  two  numbers  are  those,  whose  difference  is  7, 
and  sum  33  ?  Ans.  13  and  20. 

5.  To  divide  the  number  75  into  two  such  parts,  that 
three  times  the  greater  may  exceed  seven  times  the  less  by 
15.  Ans.  54  and  21. 

6.  In  a  mixture  of  wine  and  cider,  ^  of  the  whole  plus 
25  gallons  was  wine,  and  ^  part  minus  5  gallons  was  cider  : 
h  »w  many  gallons  were  there  of  each  1 

Ans.  85  of  wine,  and  35  of  cider. 


JCQCATION8     OF     THE     FIliST     DEGIIEK.         13\ 

7.  A  bill  of  £120  was  paid  in  guineas  and  moidores,  and 
the  number  of  pieces  used,  of  both  sorts,  was  just  100.  if 
the  guinea  be  estimated  at  21 5,  and  the  moidore  at  27^,  how 
many  pieces  were  there  of  each  sort  1         Ans.  50  of  each. 

8.  Two  travellers  set  out  at  the  same  time  from  London 
and  York,  whose  distance  apart  is  150  miles.  One  of  them 
goes  8  miles  a  day,  and  the  other  7  :  in  what  time  will  they 
meet?  Ans.  In  10  days. 

9.  At  a  certain  election,  375  persons  voted  for  two  candi- 
dates, and  the  candidate  chosen  had  a  majority  of  91 :  how 
many  voted  for  each  1 

Ans.  233  for  one,  and  142  for  the  other. 

10.  A  person  has  two  horses,  and  a  saddle  worth  £50. 
Now,  if  the  saddle  be  put  on  the  back  of  the  first  horse,  it 
will  make  his  value  double  that  of  the  second;  but  if  it  be 
put  on  the  back  of  the  second,  it  will  make  his  value  triple 
that  of  the  first.     What  is  the  value  of  each  horse] 

Ans.  One  £30,  and  the  other  £40. 

11.  The  hour  and  minute  hands  of  a  clock  are  exactly 
together  at  12  o'clock:  when  will  they  be  again  together? 

Ans.  \h.  b^j?7i. 

12.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer 
in  12  days ;  but  when  the  man  was  from  home,  it  lasted 
the  woman  30  days  :  how  many  days  would  the  man  alone 
be  in  drinking  it  ?  Ans.  20  days. 

13.  If  32  pounds  of  sea-water  contain  1  pound  of  salt, 
how  much  fresh  water  must  be  added  to  these  32  pounds, 
in  order  that  the  quantity  of  salt  contained  in  32  pounds  of 
the  new  mixture  shall  be  reduced  to  2  ounces,  or  ^  of  a 
pound  1  Ans.  224/6. 

14.  A  person  who  possessed  100,000  dollars,  placed  the 
greater  part  of  it  oui  at  5  per  cent  interest,  and  the  other 


132  ELEMENTART     ALGEBRA. 

at  4  per  cent.     The  interest  which  he  received  for  the  whole 
amounted  to  4640  dollars.     Required  the  two  parts. 

Ans.  64,000  and  36,000. 
15.  At  the  close  of  an  election,  the  successful  candidate 
had  a  majority  of  1500  votes.  Had  a  fourth  of  the  votes 
of  the  unsuccessful  candidate  been  also  given  to  him,  ho 
would  have  received  three  times  as  many  as  his  competitor, 
wanting  three  thousand  five  hundred  :  how  many  votes  did 
each  receive!  .       j  1st,  6500. 

^^^•(2d,  5000. 
15.  A  gentleman  bought  a  gold  and  a  silver  watch,  and 
a  chain  worth  $25.  When  he  put  the  chain  on  the  gold 
watch,  it  and  the  chain  became  worth  three  and  a  half  times 
more  than  the  silver  watch ;  but  when  he  put  the  chain  on 
the  silver  watch,  they  became  worth  one-half  the  gold  watch 
and  15  dollars  over :  what  was  the  value  of  each  watch  ? 

J        J  Gold  watch,  $80. 
^''^-     (Silver     "      $30. 

17.  There  is  a  certain  number  expressed  by  two  figures, 
which  figures  are  called  digits.  The  sum  of  the  digits  is 
11,  and  if  13  be  added  to  the  first  digit  the  sum  will  be  three 
times  the  second :  what  is  the  number  ?  Ans.  56. 

18.  From  a  company  of  ladies  and  gentlemen  15  ladies 

retire;    there  are  then  left  two   gentlemen  to  each  lady. 

After  which,  45  gentlemen  depart,  when  there  are  left  5 

ladies  to  each  gentleman :  how  many  were  there  of  each  at 

first  ?  A       i  ^^  gentlemen. 

'   '  (  40  ladies. 

19.  A  person  wishes  to  dispose  of  his  horse  by  lottery. 
If  he  sells  the  tickets  at  $2  each,  he  will  lose  $30  on  his 
horse  ;  but  if  he  sells  them  at  $3  each,  he  will  receive  $30 
more  than  his  horse  cost  him.  What  is  the  value  of  the 
horse  and  number  of  tickets?        ^^^    j  Horse,  $150. 

of  tickets,     60. 


A       i  Hors 
^'*^-  1No.( 


EQUATION  a     OK     THE     FIRST     DKGKEE.         133 

20.  A  person  purchases  a  lot  of  wheat  at  $1,  and  a  lot  of 
r}'e  at  75  cents  per  bushel,  the  whole  costing  him  $117,50. 
He  then  sells  -J  of  his  wheat  and  I  of  his  rye  at  the  same 
rate,  and  realizes  $27,50.     How  much  did  he  buy  of  each  ? 

.       J  80  b^ish.  of  wheat 
^"*-  I  50  bush,  of  rye. 


Equ2tions  involving  tJiree  or  more  unknown  quantities. 

77.  Let  us  now  consider  equations  involving  three  or 
more  unknown  quantities. 
Take  the  equations 

5ar  — 6y  +  42r  =  15, 
7a;-f-4y-3z  =  19, 
2x-^    y  -{-Gz  =  46. 

To  eliminate  z  by  means  of  the  first  two  equations,  mul- 
tiply  the  first  by  3  and  the  .second  by  4 ;  then,  since  the 
co-efficients  of  z  have  contrary  signs,  add  the  two  results 
together.     This  gives  a  new  equation : 

43ar  -  2y  =  121. 

Multiplying  the  second  equation  by  2,  (a  factor  of  the 
co-efficient  of  z  in  the  third  equation,)  and  adding  the  result 
with  the  third  equation,  we  have 

lGa;-i-9y  =  84. 

Tlie  question  is  then  reduced  to  finding  the  values  of  x 
and  y,  which  will  satisfy  these  new  equations. 

Now,  if  the  first  be  multiplied  by  9,  the  second  by  2,  and 
Uie  results  added  together,  we  find 

419.ir  r^  1257,     whence     z  =  3. 


134  ELEMENTARY     ALGEBR.l. 

We  might,  by  means  of  the  two  equations  involving  x 
and  y,  determine  y  in  the  same  way  that  we  have  deter- 
mined x\  but  the  value  of  y  may  be  determined  moro 
simply,  by  observing,  that  the  last  of  these  two  equations 
becomes,  by  substituting  for  x  its  value  found  above, 

48  -h  9y  =  84,   whence   y  = =  4. 

y 

In  the  same  manner,  the  first  of  the  three  picposed  equa 
tions  becomes,  by  substituting  the  values  of  x  and  y, 

24 

15  —  24  +  42  —  15,   whence   0  =  —  =  6. 

Hence,  to  solve  equations  containing  three  or  more  un- 
known quantities,  we  have  the  following 

EULE. 

I.  Eliminate  one  of  the  unknown  quantities  by  combining 
any  one  of  the  equations  with  each  of  the  others  ;  there  will 
thus  he  obtained  a  series  of  new  equations  containing  one  less 
unknown  quantity. 

II.  Eliminate  another  unknown  quantity  by  combining  one 
of  these  new  equations  with  each  of  the  others. 

III.  Continue  this  series  of  operations  until  a  single  equa 
Hon  containing  but  one  unknowti  quantity  is  obtained,  from 
which  the  value  of  this  unknown  quantity  is  easily  found. 
Then,  by  going  back  through  the  series  of  equations  that  have 
been  obtained,  the  values  of  the  other  unknown  quantities  may 
he  successively  determintd. 

77.  Give  the  general  rule  for  solving  equations  involving  three  or 
more  unknown  quantities  ?  What  is  tlie  first  step  ?  What  the  secoiu]  I 
What  the  thud  I 


EQUATIONS     OF     THE     F1H8T      DEGKEE.  135 

78.  Remark. — It  often  happens  that  each  of  the  proposed 
equations  does  not  contain  all  the  unknown  quantities.  In 
this  case,  with  a  little  address,  the  elimination  is  very 
quickly  performed. 

Take  the  four  equations  involving  four  unknown  quanti- 
ties : 

(1.)  2x-S7/  +  2z  =  13.  (3.)  4y  +  2z=  14. 

(2.)  4u  -2x  =  30.  (4.)  5y  +  3w  =  32. 

By  inspecting  these  equations,  we  see  that  the  elimination 
of  z  in  the  two  equations,  (1)  and  (3),  will  give  an  equation 
involving  x  and  y ;  and  if  we  eliminate  u  in  the  equations 
(2)  and  (4),  we  shall  obtain  a  second  equation,  involving  x 
and  y.  These  last  two  unknown  quantities  may  therefore 
be  easily  determined.  In  the  first  place,  the  elimination  of 
z  from  (1)  and  (3),  gives 

7y  -  2a:  =  1  ; 

That  of  u  from  (2)  and  (4),  gives 

20y  -\-Gx=  38. 

Multiplying  the  first  of  these  equations  by  3,  and  adding 

41y=:41; 
Whence  y  =    1. 

Substituting  this  value  in  7y  —  2x  =  ] ,  we  find 

x  =  S. 
Substituting  fo.  x  its  value  in  equation  (2),  it  becomes 

4u  -  G  zz:  30  : 
Whence  u  =  9. 

And  Rul^stituting  for  y  its  value  in  equation  (3),  there 
results 

2  =  5. 


180 


ELEMENTARY     ALUEBKA, 


1,  Given  -< 


EXAMPLES, 

x-\-    2y+    32  =  62 


to  find  a;,  y  and  j. 


2.  Giren 


^ns.  a;  =  8,  y  =  9,  2  =  12. 

2^  4-    4y  -    32  =  22  ^ 

4a;  —    2y  4-    5^  =  18   >  to  find  a;,  y  and  2. 

6a;  4-    T'y  -      2  =  63  J 

Ans.  a;  =  3,  y  =  7,  2  =  4. 


3.  Given  - 


«  +  2^  +  -^^  =  32 


>■  to  find  a;,  y  and  z. 


Ans.  a;  =  12,  y  =  20,  2  =  30. 

4.  Divide  the  number  90  into  four  such  parts  that  the 
first  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  shall  be  equal 
each  to  each. 

This  problem  may  be  easily  solved  by  introducing  a  new 
unknown  quantity. 

Let  X,  y,  2,  and  u,  be  the  required  parts,  and  designate  by 
m  the  several  equal  quantities  which  arise  from  the  condi- 
lions.     We  shall  then  have 


a;  -f  2  =  m,     y  —  2  =  M,     22  =  7/1,     ---  = 


ftu 


KyUATIONS     OF     THK     FIR  B^NljJ  \.kKt,_    ,.  187 

From  which  we  find, 

vn, 
X  —  m  —  2j  y  =  m  -\-  2,  z  =  —  y    u  =  2m, 

i 

xVnd  by  adding  the  equations, 

x-rV-f-z-fM  =  m-|-n-(-—  -f-  2m  =  4jm. 

And  since,  by  the  conditions  of  the  problem,  the  first 
member  is  equal  to  00,  we  have 

A\m  =  90     or,     f  m  =  90  ; 

hence,  m  =  20. 

Having  the  value  of  m,  we  easily  find  the  other  values ; 
viz. 

a:  =  18,     y  =  22,     z  =  10,     u  =  40. 

5.  There  are  three  ingots  composed  of  different  metals 
mixed  together.  A  pound  of  the  first  contains  7  ounces  of 
silver,  3  ounces  of  copper,  and  6  of  pewter.  A  pound  of 
the  second  contains  12  ounces  of  silver,  3  ounces  of  copper, 
and  1  of  pewter.  A  pound  of  the  third  contains  4  ounces 
of  silver,  7  ounces  of  copper,  and  5  of  pewter.  It  is  re- 
quired to  find  how  much  it  will  take  of  each  of  the  three 
ingots  to  form  a  fourth,  which  shall  contain  in  a  pound,  8 
ounces  of  silver,  3|  of  copper,  and  4}  of  pewter. 

Let  X,  y,  and  z  represent  the  number  of  ounces  which  it 

is  necessary  to  tal:e  from  the  three  ingots  respectively,  iu 

order  to  form  a  pound  of  the  required  ingot.     Since  there 

are  7  ounces  of  silver  in  a  pound,  or  10  ounces,  of  the  first 

ingot,   it  follows*   that  one  ounce  of   it  contains  ^g  of  au 

ounce  of  silver,   and   consequently    in   a  number  of  ounces 

7x 
denoted  by  z.  there  is  — -  ounces  of  silver.      In  the  same 
10 

7 


138  ELEMENTARY     ALGEBRA. 

12y  42 

manner  we  would  find  that and    —  ,  express  the  nuiu- 

bor  of  ounces  of  silver  taken  from  the  second  and  third,  to 
form  the  fourth;  but  from  the  enunciatkL.,  one  pound  of 
this  fourth  ingot  contains  8  ounces  of  si.ver.  We  have, 
then^  for  the  first  equation, 

or,  making  the  denominators  disappear, 

'7x  +  12y  +4.z=  128. 
As  respects  the  copper,  we  should  find 

3ar  -I-  3y  +  72  ==  GO, 
and  with  reference  to  the  pewter 

(jz  +  2/  -{-bzr=  G8. 

As  the  co-efficients  of  y  in  these  three  equations,  are  the 
most  simple,  it  is  convenient  to  eliminate  th's  unknown 
quantity  first. 

Multiplying  the  second  equation  by  4,  and  subtracting  the 
first  from  it,  member  from  meniber,  we  have 

bx  +240=-.  112. 

Multiplying  the  third  equation  by  3,  and  subtracting  the 
second  from  the  resulting  equation,  we  have 

15a:  +  80  =  144. 

Multiplying  this  last  equation  by  3,  and  subtracting  the 
preceding  one  from  the  resulting  equation,  y\e  obtain 

AOx  =  320, 

wbeuce  «  =  S. 


EQUATIONS     OF     TUE     FIRBT     DEGREE.         139 

Substitute  this  value  for  x  in  the  equation, 
\bx  +  8z  =  144  ; 
It  becomes  120  +  82  =  144, 

whence  «  =  3. 

Lastly,  the  two  values  a:  =  8,  z  =  3,  being  substituted  in 
tlie  equation 

6ar  +  y  -f  52r  =  68, 
give  48 -f-y-f  15  =  08, 

whence  y  =  o. 

Therefore,  in  order  to  form  a  pound  of  the  fourth  ingot, 
we  must  take  8  ounces  of  the  first,  5  ounces  of  the  second, 
md  3  of  the  third. 

Verification, 

If  there  V)e  7  ounces  of  silver  in   16  ounces  of  the  first 

in^'ol,  in  8  ounces  of  it,  there  should  be  a  number  of  ounces 

of  silver  expressed  by 

7x8 

~16- 

In  like  manner, 

1-2  X  r.  ,4X3 

— ;  .—     and     — .  .—  V 

16  16 

will  express  the  quantity  of  silver  coiitaiiiod  in  5  ounces  of 

the  serond  inj^nt,  and  3  ounces  of  the  third. 

Now,  we  have 

7x8,12x5       4x3       128       „ 

16     ^       16       ^      16  16  ' 

leTef<»re,  a  poinid  of  the  fourth    ingot  contains  8  ounces  of 

silver,  as  required  by  the  enunciation.     The  same  condition* 

may  be  venfied  with  respect  to  tlic  copper  and  pewter. 


140  ELEMENTARY     ALGEBUA. 

6.  A's  age  is  double  B's,  and  B's  is  triple  of  C's,  and  the 
Bum  of  all  their  ages  is  140.     What  is  the  age  of  each  1 

Ans.  A's  =  84,  B's  =  42,  and  C's  ==  14, 

7.  A  person  bought  a  chaise,  horse,  and  harness,  for  £G0 ; 
the  horse  came  to  twice  the  price  of  the  harness,  and  the 
chaise  to  twice  the  price  of  the  horse  and  harness.  What 
did  he  give  for  each  ?  (  £13     66*.  Sd.  for  the  horse. 

Ans.  •]  £  0   ISs.  4</.  for  the  harness. 
(  £40  for  the  chaise. 

8.  To  divide  the  number  36  into  three  such  parts  that  J 
of  the  fij-st,  ^  of  the  second,  and  \  of  the  third,  may  be  all 
equal  to  each  other.  Ans.  8,  12,  and  16. 

>^  9.  If  A  and  B  together  can  do  a  piece  of  work  in  8  days, 
A.  and  C  together  in  9  days,  and  B  and  C  in  ten  days ;  how 
many  days  would  it  take  each  to  perform  the  same  work 
alone  1  Ans.  A  14f|,  B-17jf,  C  23/1. 

10.  Three  persons,  A,  B,  and  C,  begin  to  play  together, 
having  among  them  all  $600.  At  the  end  of  the  first  game 
A  has  won  one-half  of  B's  money,  which,  added  to  his  own, 
makes  double  the  amount  B  had  at  first.  In  the  second 
game,  A  loses  and  B  wins  just  as  much  as  C  had  at  the 
beginning,  when  A  leaves  off  with  exactly  what  he  had  at 
first.     How  much  had  each  at  the  beginning'? 

y  A71S.  A  $300,  B  $200,  C  $100. 

11.  Three  persons,  A,  B,  and  C,  together  possess  $8640. 
If  B  gives  A  $400  of  his  money,  then  A  will  have  $320 
more  than  B;  but  if  B  takes  $140  of  C's  money,  then  B 
and  C  will  have  equal  sums.     How  much  has  each? 

Ans.  A  $800,  B  $1280,  C  $1560. 

12.  Three  persons  have  a  bill  to  pay,  which  neither 
ulone  is  able  to  discharge.  A  says  to  B,  "Give  me  the 
llh  of  your  money,  and  then  1  can  pay  the  bill."  B  says 
to  C,  "  Give  me  the  8th  of  yours,  and  I  can  pay  it.     Bui 


BQUATT0N6     OF     T  H  15     FIRBf     DEGREE.  141 

C  says  to  A,  "  You  must  give  me  the  half  of  yours  before 

I  can  pay  it,  as  I  liave  but  |i8."     What  was  ilm  amount  of 

their  bill,  and  how  much  money  had  A  and  R  ? 

(  Amount  of  the  bill,  $13 
Ans.  ^ 


'■\ 


A  had  $10,  and  B  ^12. 
13.  A  person  possessed  a  certain  capital,  which  he  placed 
out  at  a  certain  interest.  Another  person,  who  possessed 
10000  dollars  more  than  the  first,  and  wlio  put  out  his  capi 
tal  1  per  cent,  more  advantageously,  had  an  income  greater 
hy  800  dollars.  A  third  person,  who  possessed  15000  dol- 
lars more  than  the  first,  putting  out  his  capital  2  per  cent, 
more  advantageously,  had  an  income  greater  by  1500  dol- 
lars. Required  the  capitals  of  the  three  persons,  and  the 
rates  of  interest. 

(  Sums  at  interest,    $30000,     $40000,     45000. 
An.s. 


isums 
Kates 


lates  of  interest,  4  5  0  pr.  cl. 

14.  A  widow  receives  an  estate  of  $15000  from  her  de- 
ceased husband,  with  directions  to  divide  it  among  two  sons 
and  three  daughters,  so  that  each  son  may  receive  twice  as 
much  as  each  daughter,  and  she  herself  to  receive  $1000 
more  than  all- the  children  together.  What  was  her  share, 
and  what  the  share  of  each  child  1 

(  The  widow's  share,   $8000. 

Ans.    }  Each  son's,  2000. 

(  Each  daughter's,  1000. 

15.  A  certain  sum  of  money  is  to  be  divided  between 
three  persons,  A,  B,  and  C.  A  is  to  receive  $3000  less 
than  half  of  it,  B  $1000  less  tiian  one-third  part,  and  C  to 
receive  $800  more  than  the  fourth  part  of  the  whole.  What 
is  the  sum  to  be  divided,  and  what  does  each  receive  ? 

Sum,  $38400. 

.         .    A  receives     1G200 

^""^B        "  11800. 

C         "  10400 


142 


ELEMENTARY     ALGEBRA, 


CHAPTER  IV. 

Of  Powers. 

79.  Tf  a  (]uantlt}  be  multiplied  any  number  of  times  b) 
its^'.lf,  the  product  is  called  a  power  of  the  quantity.     Tlius, 

a  —  a^  is  a  root,  or  first  power  of  a. 

a  X  f>  =  (i^  is  the  square,  or  second  power  of  a. 

a  X  a  X  «  =  w^  is  tlie  cuhe,  or  third  power  of  a. 

a  X  a  X  a  X  a  —  a*  is  the  fourth  power  of  a. 

aXaxaxaxu  —  a^  is  the  fifth  power  of  a. 

In  every  power  there  are  three  things  to  be  considered  * 

Is^.  The  quantity  which  is  multiplied  by  itself,  and  which 
is  called  the  ruot^  or  the  first  power. 

2d.  The  small  figure  which  is  placed  at  the  right,  and  a 
little  above  the  letter.  This  figure  is  called  the  exponent 
of  the  power,  and  shows  how  many  times  the  letter  enters 
as  a  factor. 

3d.  The  power  itself,  which  is  the  final  product,  or  result 
of  the  multiplications. 

71'.  Tf  a  quantity  be  continually  multiplied  by  itself,  what  is  the  pro- 
juct  called  ?  jiow  many  things  are  to  be  cousidered  in  every  power  H 
What  are  thev 


OF    POWERS.  143 

For  example,  if  we  suppose   a  =  3.;  we  have 

a  =z      3  the  1st  power  of  3. 

a2  =  32  =  3x3=      9  the  2d  j)uwer  of  3. 

o'  =  33  =  3    X  3  X  3  =    27  the  3d  power  of  3. 

a<  =  3*  =  3    X  3    X  3  X  3  =    81    the  4th  power  of  3. 

a»  =  3*  =  3    X  3    X  3    X  3  X  3  =  243  the  5th  power  of  3. 

In  thj'se  expressions,  3  is  the  root,  1,  2,  3,  4  and  5  are 
tlie  exponents,  and  3,  9,  27,  81   and  243  are  the  powers. 

To  raise  a  monomial  to  any  power. 

80.  Let  it  be  reqjiired  to  raise  the  monomial  2aW  to 
the  fuurth  ])uwer.     We  have 

(2«  '62)*  =  2a^62  X  2aW  X  2a^62  X  2«^6^ 

which  nuMcly  expresses  that  the  fourth  power  is  equal  to 
the  pn»<liu'i  which  arises  fmru  taking  the  quantity  four 
times  as  a  factor.  By  the  rules  for  multiplication,  this  pro- 
duct is 

{^aWy  =  2*a''  +  34-3  + 3^2  +  2  +  2+ 2  _  2*a^%^  ; 
from  which  we  see, 

1st.  That  the  co-efficient  2  must  be  raised  to  the  4th 
p<.>wer  ;    and, 

2d.  That  the  exponent  of  each  letter  must  be  multiplici 
by  4,  the  exponent  of  the  pow(!r. 

^<  the  same  reasonina  wuiijtl  applv  to  every  exani[»lo, 
wt-  ha\e,  for  ihc  iai:>inj;  ol  munoiiiiais  Lo  any  power,  the 
follow  ing 


'w: 


O"^ 

w 


144  ELEMENTARY     ALGEBRA-. 

RULE. 

J.  Raise  the  co-efficient  to  the  required  power, 
II.  Multiply  the  exponent  of  each  letter  by  the  expomnt  c/ 
Qve  power. 

EXAMPLES. 

1.  What  is  the  square  of  oa^y'^l  Ans.  \)a*y^ 

2.  What  is  the  cube  of  Ga^y^ar  ?  Ans.  2lQa^^y^x'' 

3.  What  is  the  fourth  power  of  2a^y^b^  ? 

Ans.    lCal2yl2^,2( 

4  What  is  the  square  of  a'^-b^y^  1  Ans.  a^b^^y* 

5.  What  is  the  seventh  power  of  a^bcd^  1 

Ans.   a^'^b'c'^d'^' 

6.  What  is  the  sixth  power  of  a'^bh'^dl       Ans.  a'^H^^c^'^d 

7.  What  is  the  square  and  cube  of   —  2a%^  1 

Square.  Cube. 

-  2a^^  -  2aW 

-  2fP  -  2«262 
-h  '4a*b*.  +  4i^6* 

By  observing  the  way  in  which  the  powers  are  formed, 
we  may  conclude, 

1st.   When  the  root  is  positive,  all  the  p(rwers  will  he  positive. 

2d.  When  the  root  is  negative^  all  powers  denoted  hy  an 
even  exponent  will  be  j)ositive,  and  all  denoted  by  an  odd  ex- 
ponent will  be  negative. 

80i  What  is  a  monomial?  Give  the  rule  for  raising  a  monomial  to  any 
power.  When  the  root  is  positive,  how  will  the  powers  be  ?  When  the 
WKit  is  negative,  how  will  the  powers  be  I 

# 


OF    POWERS.  145 

a  What  is  the  square  of   —  2  a^6*  ?  Ans.  -ia^b^^ 

9.  What  is  the  cube  of   —bw'yHI    Ans.   —  l*25a'*yV. 

10.  What  is  the  eighth  power  of   —  a?xy^  \ 

Ans.    -\-  a^^jc^y^^ 

11.  What  is  the  seventh  power  of   —  a^yx"^  1 

Ans.    —  a^^y'x^* 

12.  What  is  the  sixth  power  of  2a6V  ^ 

Ans.  QAa^-^^y^^ 

13.  What  is  the  ninth  power  of   —  cdxhj^l 

Ans.    —  c^d^x^^y'^'' 

14.  What  is  the  sixth  power  of   —  ZalHI 

Ans.  72da%^^d\ 

15.  What  is  the  square  of   -  lOa^AV  ?     Ans.  100a*6*c«. 

16.  What  is  the  cube  of   -  da%'>dy^  1 

Ans.    -  729ai86»Vy«. 

17.  What  is  the  fourth  power  of   —  4a*6Vt/*  ? 

Ans.  256a2«6i2c'«if^. 

18.  What  is  the  cube  of   —  4a^^c^dl 

Ans.    —G4a%^c^d^. 

19.  What  is  the  fifth  power  of  2a^b^xy  1 

Ans.  32a'56io^y^ 

20.  What  is  the  square  of  20ar^yV?         Ans.  400xYc^\ 

21.  What  is  the  fourth  power  of  Sa^ftV? 

Ans.  Sla%^c^\ 

22.  What  is  the  fifth  power  of  —  cWx^y^  1 

Ans.     —  c  Vi*x"^yi*' 
2.3,  What  is  the  sixth  power  of    —  uc^dfl 

Ans.  a^c^'^dy 

24.  What  is  the  fourth  power  of   -  2aVci^  1 

Ans.  lGa8c«(/i2. 


146  ELEMENTARY      A.LGSBRA. 


To  raise  a  polynomial  to  any  power. 

S).  A  power  of  a  polynomial,  like  that  of  a  monomial, 
h  obtained  by  multiplying  the  quantity  continually  by 
itself.  Thus,  to  find  the  fifth  power  of  the  binomial  a  +  6, 
wo.  have 

a   -{-    b 1st  power. 

a    -f-    b 
a*  ^    ab 

-i-    ab  +  b^ 

-\-  2ub  -i-  b^        2d  power. 


a2  -f  2ub  -f  62 

a   -\-    b  

aM^a26+      ^ 

a3  +  3^26  4-    3a62   H-    6^    ....     3d  power. 

a   -\-    b 

a*  +-  3a-^6  +    Su^o^  +    a63 
.  +    aH>  +    3tf-^62  _^    3^,^,3   ^  ^,4 

a^  -f  4a^6  -f-    Ga'^62  -|-  4a63   -|-    6*        4th  power. 

a    +   6 

a'   \-4<r^b~+~Qn^l^  -{-  4a%^  -{-    ab* 

-(-    «H  4-    4a '^62  -f-  (JaV)3  -f-  4«6'^  +  6^ 

a*  +  5a^6  +  lOa-^62  -|-  10a263  -f-  5a6^  +  6^     J/i5. 

Remark. — 82.  It  will  be  observed  that  the  number  of 
multiplications  is  always  1  less  than  the  units  in  the  expo- 


81    llow  is  thtt  power  of  a  polynomial  obtained  I 


or    POWERS.  147 

«.ent  of  the  pov.er.  Thus,  if  the  exponent  is  1,  no  multipli- 
wition  Is  nect'ssaiy.  it'  it  is  2,  v\e  multiply  once  ;  if  it  is  3, 
twice;  if  4,  three  times,  &c.  The  powers  of  polyiioinials 
may  bt  e.\ pressed  by  means  of  an  exponent.  Thus,  to  e\- 
presfc  that  a  H-  6  is  to  be  raised  to  the  5th  power,  we  write 

(a  +  by, 

which  is  the  expression  for  the  fifth  power  of  a  +  6. 
2.  Find  the  5th  power  of  the  binomial  a  —  b. 

a   —    h         lat  power. 

a   ^    b 

Qp-  —    ab 

"    q6   -f  6» 

a2-2</6   -1-62 2d   power. 

a    -      b 

a^  —  ^a'^b  -h      a6» 

~    a^-\-    2ub^    -y 
a^  -  Sa^b  -j-    3a62   _  63     .     .     .     .      3d   power. 

g   -    6 

a*  —  3a  ^6  -f-    3aH^  —      ab^ 

-  a^b  +    3«^62  -    3a63   -f-   6* 

a*  —  4t/ '6  4-    6a-^62  —    4a6^   -f   6*        4th  power. 

-  6 


a»_4a*6-f-    (k|362  —    4a'^63  _^    «^4 

—    a*M^4a'^62—    6a^63  -(-  4«6*  --  6^ 
a»  -  5u*6  -I-  rOa^'62ir"l0a'^6H-^a6^i^~6^ 


-4/jj. 


iJ2.  How  <^i^er^  the  number  of  multiplications  cnmpare  \rifli  (hp 
t.v|K)Ui-ul  of  the  power  I  If  the  expuueut  is  4,  bow  muuy  luuliipUca 
tiaiiA  t 


148  ELEMENTARY     ALGEBRA. 

3.  What  is  the  square  of    5a  —  2c  -\-  d  'i 

5a    —    2c    +      d 
5a    —    2c    +      d 


25a2  —  lOac  +    5ad 

—  IQac  +    4c2  —  2cd 

+    bad  —  2cd-\-    d'^ 
25a^  —  20«cTT0«c^  -f  4c2  -  4crf  -}-  J^     ^1^. 


4 ,  Find  the  4th  power  of  the  binomial   3a  ■—  2h. 

3a  —  26        .......       1st  power, 

3a   -  2h 

9a2  —  6a6 

—  6a6   4-46 


9a2  —    12«6   4-  462     .     ,     .     .     .       2d  power. 
3a   -      26 


27a3  _    3(;a2^  _^  i2a62 

-    18a^6  4-24«6^-    86^ 

27a3  _  ~54a26~+36a62  —    86^     .  3d  power. 

3a    -      26 

81a*  —  lG2a36  +  1080^62  _  24a63 

—    54a36  4-  108«262  —  72a63  +  166* 
81a*  — TlGa"^6"T2H)«^62  —  96a63  ^  ig6* 

5.  What  is  the  square  of  the  binomial  a  4-  1"? 

An%,  a2  +  2a  -f  J 

6.  What  is  the  square  of  the  binomial   a  —  1  ? 

A^s.  a2  —  2a  -f  1, 

7.  What  is  the  cube  of  9a  —  36  ? 

An8.  729a3  _  729a26  +  243a62  —  276'. 

8.  Wliat  is  the  third  power  of   a  —  1  ? 

Ans.  a»  —  3a2  -f  3a  -  i. 


OF      POWERS.  149 

9    What  is  the  4th  power  of  x  —  y^ 

Ans.  x^  —  Axr^y  -\-  6x^y^  —  4xy^  4-  y*. 

10    What  is  the  cube  of  the  trinomial   x  -\-  y  ■\-  z'^ 

Answer. 
«M  3j:2>  -|-3a:22r-h3a'y2-^3a-02-^3j/22-|-3yz2-h6jryz  +  y3.f  ^3. 

1 1 .  What  is  the  cube  of  the  trinomial    2a?  —  4a6  -f-  36^  1 
Aiisiver. 
8a   -  48a*6  +  132a*62  _  208a363+  198a26*-108a6*-f  276« 

To  raise  a  fraction  to  any  power. 

83.  A  power  of  a  fraction  is  obtained  by  multiplying  the 
fraction  by  itself  a  certain  number  of  times ;  that  is,  by 
multiplying  the  numerator  by  the  numerator,  and  the  deno- 
minator by  the  denominator. 

Thus,  the  cube  of   —     which   is  written 


(a  \  3        a         a         a         a 


Is  found  by  cubing  the  numerator  and  denominator  sepa- 

rately. 

a  —  c 
2.  What  is  the  square  of  the  fraction   -j—r —  1 

We  have 

/a-cY_{a-cY_  a^-2ac  +  c^ 


.3«/3 


xy  A  x^y 

3.  What  is  the  cube  of   -r—  »  Ans.  T^^rr  i 

36c  21b^(r 


81    Uuw  do  yuu  find  Um  power  of  a  frmctioo  I 


"l'^>0  EtEMENTART     ALGEBRA 

4.   What  is  the  fourth  power  of  -  —  ? 


6.  What  is  the  cube  of ^  ? 

X  +  y 

6.  W^hat  is  the  fourth  power  of 1         Ans. 


7.  What  is  the  fifth  power  of    — —  1        Ans. 


lSy2    *  32y^£^ 


8.  What  is  the  square  of  % ] 

by  —  X 


•^"^-     h2y2  _  2bxy  -h  x^  ' 

9.   What  is  the  cube  of   ?^^^-^  2 

X  -\-2y' 

8a^  -  S6a^  +  54ab^  -  2763 
^^'    x^  4-  6a:V  +  l^^i/^  +  8y^  * 

Binomial  Formula, 

84.  The  method  which  has  been  explained  of  raising  a 
binomial  to  any  power,  is  somewhat  tedious,  and  hence 
other  methods,  less  diflicult,  have  been  anxiously  sought 
for.  The  most  simple  which  has  yet  been  discovered,  is 
that  of  Sir  Isaac  Newton,  by  means  of  the  Minumial 
Formula. 


84.  What  is  the  object  of  the  Bbiouiial  Formula?     Who  diacovtrcU 
tiiia  funuula  t 


BINOMIAL    FORMULA.  151 

85.  In  raising  a  quantity  to  any  power,  it  is  plain  that 
there  are  four  things  to  be  considered : — 

1st.    Tlie  number  of  terms  of  the  power, 
2*1    The  signs  of  the  terms. 
3d,    The  exponents  of  the  letters. 
4th-   The  co-eilicieiits  of  the  terms. 

Of  the  Terms, 

86.  If  we  take  the  two  examples  of  Article  81,  which 
we  there  wrought  out  in  full ;  we  have 

(a  H    by  =  a*  -f  5«<6  -|-  lOa-^^^  -f  lOt/263  4.  ^ab*  -(-  6*  ; 

(a  -  by  =  a*  -  ba*b  -f-  lOa'b^  -  iOuH^  -h  bub'  -  h\ 

By  exnmining  the  several  multiplications,  in  Art.  81,  we 
shall  observe  thut  the  second  power  uf  a  binomial  contains 
three  terms,  the  third  power  four,  the  fourth  power  five,  the 
fifth  power  six,  Aic. ;  and  hence,  we  may  conclude — 

Tkut  the  number  of  lenius  in  any  power  of  a  binomial^  it 
greater  by  one,  than  the  exponent  of  the  2)0wer. 

Of  the  Sif/ns  of  the  Terrns. 

87.  It  ib  evident  that  when  both  terms  of  the  given 
binomial  are  plus,  all  the  terms  of  the  power  will  be  plus, 

2d.  If  the  second  term  of  the  binomial  is  negative,  then 
all  the  odd  terniH,  counted  from  the  left,  will  be  positive,  and 
all  the  even  terms  negative. 

85.  In  rai:<ing  a  quantity  to  any  power,  how  many  things  are  to  be 
cnn>iiliTe«l  ?     Wliat  are  tliey  ? 

8U.  H(iw  many  terms  are  there  in  any  power  of  a  binomial !  If  the 
ex|)onent  m  3.  how  many  terms!     If  it  i8  4,  h«»w  many  terms  f    If  6?  dc 

87.  If  b«>tlj  terms  of  the  binomial  are  positive,  how  are  the  tertnM  (if 
the  power  f  if  the  »ccond  term  is  negative,  Low  are  thefii^jn*  of  ILm 
tenust 


152  ELEMENTARY     ALGEBRA. 

Of  the  Exponents. 

88.  The  letter  which  occupies  the  first  place  n  a  bino 
mial,  is  called  the  leading  letter.  Thus,  a  is  the  leading 
letter  in  the  binomials  a  +  5,  a  —  h. 

1st.  It  is  evident  that  the  exponent  of  the  leading  letter, 
id  the  first  term,  will  be  the  same  as  the  exponent  of  the 
power;  and  that  this  exponent  will  diminish  by  unity  in 
each  term  to  the  right,  until  we  reach  the  last  teim,  which 
does  not  contain  the  leading  letter. 

2d.  The  exponent  of  the  second  letter  is  1,  in  the  second 
term,  and  increases  by  unity  in  each  term  to  the  right  until 
we  reach  the  last  term,  in  which  the  exponent  is  the  same 
as  that  of  the  given  power. 

3d.  The  sum  of  the  exponents  of  the  two  letters,  in  any 
term,  is  equal  to  the  exponent  of  the  given  power.  This 
last  remark  will  enable  us  to  verify  any  result  obtained  by 
means  of  the  binomial  formula. 

Let  us  now  apply  these  principles  in  the  two  following 
examples,  in  which  the  co-efficients  are  omitted : — 

(a  +  &)6  .  .  .  a^  ^-jCL^h  -f  0*62  +  aW  +  a^-h^  +  ah^  +  ^'^ 

(a  —  6)6  ...  a6  —  w>h  +  af'lP-  —  aW  +  a^h'^  —  ah^  -f  h^. 

As  the  pupil  should  be  practised  in  writing  the  terms, 
with  their  proper  signs,  without  the  co-efficients,  we  will  add 
a  few  more  examples. 

88.  Which  is  the  leading  letter  of  a  hinomial  ?  "What  is  the  exponent 
of  this  letter  in  the  first  term?  How  djes  it  change  in  the  terms  to 
wards  the  right?  What  is  the  exponent  of  the  second  letter  in  the 
second  term  ?  How  does  it  change  in  the  terms  towards  the  right  \ 
What  is  it  in  the  last  term  ?  What  is  the  sum  of  the  exponents  in  au) 
tenu  equal  to  ? 


BINOMIAL     FORMULA..  153 

1.  (a  -f-  &)s  .  .  a3  -+-  a%  -f  ah^  -\-  P. 

2.  (a  -  6)*  .  .a*  -  a^  +  a^t^  -  06^  -f  6». 

3.  (a  -H  6)5  .  .  a*  +  a*b  +  a^^s  4.  ^2^3  _|_  ^6*  +  ^s. 

4.  [a -by.  .  a'-a%-\-a^b^-a*P-{-a^h*-a^^-}-ab*—b\ 

Of  the  Co-efficients. 

89.  Tlic  00-efficient  of  the  first  term  is  unity.  The  co 
etficient  of  the  second  term  is  the  same  as  the  exponent  of 
the  given  power.  The  co-efficient  of  the  third  term  is  found 
by  multiplying  the  co-efficient  of  the  second  term  by  the 
exponent  of  the  leading  letter,  and  dividing  the  product  by 
2.     And  finally— 

If  the  co-ejiciefit  of  any  term  be  multiplied  by  the  exponent 
of  the  leading  letter,  and  the  product  divided  by  the  number 
which  marks  the  place  of  that  term  from  the  left,  the  quotient 
will  be  the  co-ejicient  of  the  next  term. 

Thus,  to  find  the  co-efficients  in  the  example 

(a-by  .  .  .  c?  -  a%  +  a^'b^-  a*b^-{-  a^b*-  a^b^-\-  ab^  -  6' 

we  first  place  the  exponent  7  as  a  co-efficient  of  the  second 
term.  Then,  to  find  the  co-efficient  of  the  third  term,  we 
multiply  7  by  6,  the  exponent  of  a,  and  divide  by  2.  The 
quotient  21  is  the  co-efficient  of  the  third  term.  To  find  the 
co-efficient  of  the  fourth,  we  multiply  21  by  5,  and  divide 
the  product  by  3:  this  gives  35.  To  find  the  co-cfiicient  of 
the  fifth  term,  we  multiply  35  by  4,  and  divide  the  product 
by  4  :  this  gives  35.  The  co-efficient  of  the  sixth  terir., 
found  in  the  same  way,  is  21  ;  that  of  the  seventh,  7  ;  and 
that  of  the  eighth,  1.     Collecting  these  co-efficients, 

(a-by  = 

a'_7afi6-f21a*62-.^,5a«63-f.35a36«_21a26*-f7a6»-A^ 
7* 


154  ELEMENTARY     ALGEBRA. 

JiEMARK. — We  see,  in  examining  this  last  result,  that  the 
co-efficients  of  the  extreme  terms  are  each  unity,  and  that 
the  co-efficients  of  terms  equally  distant  from  the  extren:e 
terms  are  equal.  It  will,  therefore,  be  sufficient  to  find  the 
co-efficients  of  the  first  half  of  the  terms,  and  from  these 
the  others  may  be  immediately  written. 

EXAMPLES. 

1.  Find  the  fourth  power  of  a  -f-  ^. 

Ans.  a*  4-  4a'^6  4-  (^a%^  +  4aP  4-  6*. 

2.  Find  the  fourth  power  of  a  —  b. 

Ans.  a*  —  Aa'b  -f  Q>aW  —  4aP  -|-  ¥. 

3.  Find  the  fifth  power  of  a  -f-  b. 

4.  Find  the  fifth  power  of  a  —  h. 

Ans.  a'^  -  5«^6  -I-  10aV>2  _  iQ^^2p  ^  5e^4  _  ^s^ 

5.  Find  the  sixth  power  of  a  -\-  h. 

Ans.  a«  -h  Ca''6  +  Iba'b'^  -f  20cr^b^  +  lbu%*  -f  Ga^**  -^  b\ 

6.  Find  the  sixth  power  of  a  —  b. 

Ans.  a^  -  6a^b  -{-  Xba'b'^  —  20a^b^  4  '^5«2M  —  Gab^  +  6« 

7.  Let  it  be  required  to  raise  the  binomial  Sa^c  —  2bd  to 
he  fourth  power. 

It  frequently  occurs  that  the  terms  of  the  binomial  are 
affected   with  co-efficients  and   exponents,  as  in   the  above 


81).  What  is  the  co-eflicient  of  tlie  first  term  ?  Wliat  is  tlu^  ro-ef^clp.it 
of  th»^  spcond  ?  How  <\o  you  find  tlu*  co-rfTiricnl  of  f|ie  ttm-il  U-imi' 
How  do  you  find  th*>  co-effici<Mit  of  iiny  triiii?  \\'li;it  mic  ihi'rofili 
cienta  of  the  first  and  last  terms?  How  are  the  co-efficieuts  of  lermfl 
equally  distant  from  the  two  extrciues  ? 


BINOMIAL      FORMULA.  155 

example.     In  the  first  place,  we  represent  each  term  of  the 
biuoniial  by  a  single  letter.     Thus,  we  place 

Sa^c  =  ar,     and     —  2bd  =  y, 

we  then  have 

{x  -f  y)*  =  x*  -\-  Ax^y  +  Ox'y^  +  4xy3  -(.  y4. 

But,       a:2  =  9aV2,        a-3  =  27aV,        a-*  =  81aV  ; 

and        y2  _  452^/2^    y^  =  _  5^,3^3^        ^4  _  iC6*t/*. 

Substituting  for  x   and   y   their  values,  we  have 

-f-  4(3«-c)  (  -  Udf  +  (  -  2&</)*, 

and  by  performing  the  operations  indicated, 

(3u^c  -  Uiiy  =  8l(/V-21f)a«c3W-f  21Ctt*c262</2_9Ga2c6V 
-I-  106V*. 

8.  What  is  the  square  of  Sa  —  Ght 

Ans.  da^  —  3Cmb  -\-  SC>b^ 

9.  What  is  the  cube  of  ^x  —  (Sy  1 

Ans.  21  x^  -  WZx'^y  -f  324ary2  _  21Gy^ 

10.  AVhat  is  the  square  of   x  —  yl 

Ans.  x"^  —  2ry  -{-  y'. 

11.  What  is  the  eighth  power  of  m  -\-  n% 

Ans.  m^ H-  Sm'n  -{- 2Sni^H^ -f-  bGtn^n^ -f- 70m*n* -f- ^Gm^n* 
+  28m2/i«  +  8mw'  +  Ji\ 

12.  What  is  the  foui  th  power  of   a  —  36  ? 

Ans.  a*  -  12a36  -|-  540^62  _  |08a6''  +  816" 

13.  What  is  the  fifth  power  of   c  —  2dl. 

Ans.  c*  -  \o/d  -f-  40f  V2  -  80cV3  -f  SOcd*  -  S2d' 

14.  Wliat  U  the  cube  of    (m  —  3t/? 

Ans.  125a3  ~  225^2 J  -f  135at/2  -  27</». 


156  ELEMENTARYALGEBRA. 

Eemark,  The  powers  of  a  polynomial  may  easily  be 
found  by  the  Binomial  Formula. 

15.  For  example,  raise   a  -^  b  -{-  c   to  the  third  power. 

First,  put     .     .     .     .     b  -\-  c  =  d: 
Then.,     {a -\- b  +  cY  =  (a -{-  df  =  a^  +  Za^d  +  Zad"^  +  #, 
Or,  by  substituting  for  the  value  of  o?, 

(a  +  ^  +  c)3  =  a3  4-  Za%  +  Zab"^  +  ^3 

Sa^c  H-  362c  4-  6«6c 

+  3ac2  +  .36c2 

+      c\ 

This  expression  is  composed  of  the  sum  of  the  cubes  of  the 
three  terms,  2:>lus  three  times  the  square  of  each  term  by  the 
product  of  the  first  powers  of  the  two  others,  plus  six  times 
the  product  of  the  three  terms.  It  is  easily  proved  that  this 
law  is  true  for  any  polynomial. 

To  apply  the  preceding  formula  to  the  development  of 
the  cube  of  a  trinomial,  in  which  the  terms  are  affected  with 
co-efficients  and  exponents,  designate  each  term  by  a  single 
letter,  then  replace  the  letters  introduced,  by  their  values,  and 
perform  the  operations  indicated. 

From  this  rule,  we  find  that 

(2a2  _  4a6  +  362)3  ^  g^e  _  48a5j  -|_  I32a462  -  ^OSa'^b^ 
+  198a264  -  108«65  +  2766. 

The  fourth,  fifth,  &c.,  powers  of  any  polynomial  can  be 
found  in  a  similar  manner. 

10.  What  is  the  cube  of  a  —  26  -f  c  ? 
Ans.  a^  —  Sb''  +  c3  -  0a26  4"  Sa^c  -f  12a62  -|-  \2bh+'^(w^ 
-  66c2  —  I2abc. 


KXTKACTION  OF  THE  SQUARE  ROOT.    157 


CHAPTER   V. 

Extrdctimi  of  tJie  Square  Boot  of  Numbers.  Fcrmation 
of  the  Square  and  Extraction  of  the  Square  Root  of 
Algebraic  Quantities.  Calculus  of  Radicals  of  tlie 
Second  Degree. 

90.  The  square  or  second  power  of  a  number,  is  the  pro- 
duct which  arises  from  multiplying  that  number  by  itself 
once:  for  example,  49  is  the  square  of  7,  and  144  is  tho 
square  of  12. 

91.  The  square  root  of  a  nutiiber  is  that  number  which, 
being  multiplied  by  itself  once,  will  produce  the  given  num- 
ber.  ITius,  7  is  the  square  root  of  49,  and  12  the  square 
root  of  144  :  for  7,  X  7  =  49,  and  12  X  12  -  144. 

92.  The  square  of  a  number,  either  entire  or  fractional,  is 
easily  found,  being  always  obtained  by  multiplying  this 
number  by  itself  once.  The  extraction  of  the  square  root 
of  a  number  is,  however,  attended  with  some  difficulty,  and 

equires  particular  explanation. 


90    What  is  the  i^quarc,  or  second  power  of  a  numher! 
HI.   Wliat  is  UiC  Hjuare  ro<>t  of  a  number! 


15S  ELEMENTARY     ALGERRA. 

The  first  ten  numbers  are, 

1,       2,       3,       4,       5,       G,       7,       8,       9,      10; 

and  their  squares, 

1,       4,       9,     10,     25,     36,     49,     64,     81,    100; 

and  reciprocally,  the  numbers  of  the  first  line  are  the  squaro 
roots  of  the  corresponding  numbers  of  the  second.  We  may 
also  remark  that,  the  square  of  a  number  exprenfied  by  a  slnyl.e 
fgure^  will  contain  no  unit  of  a  higher  denoiidnation  than 
tens.* 

The  numbers  of  the  last  line,  1,  4,  9,  16,  &c.,  and  all 
other  numbers  which  can  be  produced  by  the  multiplication 
of  a  number  by  itself,  are  called  2)erfect  squares. 

It  is  obvious  that  there  are  but  nine  perfect  squares  among 
all  the  numbers  which  can  be  expressed  by  one  or  two  figures : 
the  square  roots  of  all  other  numliers  expressed  by  one  or 
two  figures,  will  be  found  between  two  whole  numbers  dif 
fering  from  each  other  by  unity.  Thus  55,  which  is  comprised 
betw^ecn  49  and  64,  has  for  its  square  root  a  number  between 
7  and  8.  Also  91,  which  is  comprised  between  81  and  100, 
has  for  its  square  root  a  number  between  9  and  10. 

93.  Every  number  may  be  regarded  as  made  up  of  a 
certain  numlier  of  tens  and  a  certain  number  of  units.  Thus 
64  is  made  up  of  6  tens  and  4  units,  and  ma}  be  ex])iessed 
under  the  form  60  -h  4. 


92i  What  will  be  the  highest  denomination  of  the  square  of  a  n  itnbei 
expressed  by  a  single  tigure  ?  What  are  perfect  tiquares  ?  Eow  m?u)> 
are  there  between  1  and  100?     What  are  the}  ? 

♦  Se«  Arithmelic.  Art.  8. 


EXTRACTION     OF     T  II  E    8  Q  U  A  R  B     ROOT.         159 

Now,  if  we  represent  tlie  tens  by  a  and  the  units  by  6, 
we  shall  have 

a  -}-  6    =  0)4, 
and  (a-f^)2=(04)2; 

or  a2  +  2tf6  +  b^  =  4000. 

Which  proves  that  the  square  of  a  number  composed  of 
tens  and  units,  equals  the  square  of  tlie  teas  plus  twice  tli€ 
product  of  tlie  tens  by  the  units,  plus  the  square  of  the  units. 

94.  If  now,  we  make  the  units  1,  2,  3,  4,  6cc.,  tens,  or 
units  of  the  second  order,  by  annexing  to  each  figure  a 
cipher,  we  shall  have 

10,     20,     30,     40,     50,     GO,     70,     80,     90,     100, 

and  for  their  squares, 

100,  400,  900,  1000,  2500,  3000,  4000,  0400,  8100,  10000. 

From  which  we  see  that  the  square  of  one  ten  is  100,  the 
square  of  two  tens  400 ;  and  generally  (hat  the  square  of 
tens  will  contain  no  unit  of  a  less  denorni/iation  than  hurt- 
drtds,  nor  of  a  hiyher  name  than  thousands. 

Ex.  1. — To  extract  the  square  root  of  0084. 

Since  this  number  is  composed  of  mure  than 
two    places   of  figures,  its    root    will    contain  00  84 

mori't   than    one.      But  since    it   is    less    than 
10000,  which  is  the  square  of  100,  the  root  will  contain  but 
two  figures:  that  is,  units  and  tens. 

Now,  the  square  of  the  tens  must  be  found   in  the  two 


98.  How  may  every  number  be  regarded  as  mad«>  no  f  What  is  ilie 
bqiiac  of  a  nuuiber  oom|H>std  of  tens  and  unit?  equal  to  f 

U4  Wliat  is  tlie  bquore  uf  oue  ten  eqiud  tut  Of  2  tou«*  0/  S 
icosf  «&c 


118  4 
118  4 


160  ELEMENTARY     ALGEBRA. 

left-hand  figures,  which  we  will  separate  from  the  other  two 
by  putting  a  point  over  the. place  of  units,  and  a  second  over 
the  place  of  hundreds.  These  parts,  of  two  figures  each,  are 
called  periods.  The  part  60  is  comprised  between  the  two 
squares  49  and  64,  of  which  the  roots  are  7  and  8  :  hence^ 
7  expresses  the  number  of  tens  sought;  and  the  required  root 
is  composed  of  7  tens  and  a  certain  number  of  units. 

The  figure  7  being  found,  we 
write  it  on  the  right  of  the  given  60  84   78 

number,  from  which  we  separate  49 

it  by  a  vertical  line  :    then  we     7  x  2  =  14 
subtract  its  square,  49,  from  60, 
which  leaves  a  remainder  of  11,  0 

to  which  we  bring  down  the  two 

next  figures  84.  The  result  of  this  operation,  1184,  con- 
tains twice  the  product  of  the  tens  by  the  units,  plus  the  square 
of  the  units. 

But  since  tens  multiplied  by  units  cannot  give  a  product 
of  a  less  unit  than  tens,  it  follows  that  the  last  figure,  4, 
can  form  no  part  of  the  double  product  of  the  tens  by  the 
units  :  this  double  product  is  therefore  found  in  the  part  118, 
which  we  separate  from  the  units'  place,  4. 

Now  if  we  double  the  tens,  which  gives  14,  and  then  divide 
118  by  14,  the  quotient  8  loill  express  the  units,  or  a  num- 
ber greater  than  the  units.  This  quotient  can  never  be  too 
small,  since  the  part  118  will  be  at  least  equal  to  twice  the 
product  of  the  tens  by  the  units :  but  it  may  be  too  large  ; 
for  the  118,  besides  the  double  product  of  the  tens  by  the 
units,  may  likewise  contain  tens  arising  from  the  square 
of  the  units.  To  ascertain  if  the  quotient  8  expresses  the 
number  of  units,  we  write  the  8  on  the  right  of  the  14, 
which  gives  148,  and  then  we  multiply  148  by  8.  Thus^ 
W'C  evidently  form,  1st,  the  square  of  thf3  units  ;  and, 
2d,   the  double   product  of  the  tens  by  the   units.     This 


JtXTRACTION     OF     THE     8QUAUE     ROOT.  161 

multiplication  being  effected,  gives  for  a  product  1184,  a 
number  equal  to  the  result  of  the  first  operation.  Having 
subtracted  the  product,  we  find  the  remainder  equal  to  0 : 
bftnce,78  is  the  root  required. 

Indeed,  in  the  operations,  we  have  merely  subtracted 
from  the  given  number  (3084,  1st,  the  square  of  7  tens,  or  of 
70;  2d,  twice  the  product  of  70  by  8 ;  and,  3d,  the  square 
of  8 :  that  is,  the  three  parts  which  enter  into  the  composi- 
don  of  the  square  70  +  8,  or  78 ;  and  since  the  result  of 
the  subtraction  is  0,  it  follows  that  78  is  the  square  root  of 
6084. 

95.  Remark. — llie  operations  in  the  last  example  have 
been  performed  on  but  two  periods,  but  it  is  plain  that  the 
same  methods  of  reasoning  are  equally  applicable  to  larger 
numbers,  for  by  changing  the  order  of  the  units,  we  do  not 
change  the  relation  in  which  they  stand  to  each  other. 

Thus,  in  the  number  (50  84  95,  the  two  periods  60  84 
have  the  same  relation  to  each  other  as  in  the  number 
60  84 ;  and  hence  the  methods  used  in  the  last  example 
are  equally  applicable  to  larger  numbers. 

96.  Hence,  for  the  extraction  of  the  square  root  of 
Qurabers,  we  have  the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  two  figures 
each^  beginning  at  the  right  hand: — the  period  on  the  Uft  will 
ojten  contain  but  one  figure. 

II.  J^nd  the  greatest  square  in  the  first  period  on  the  left^ 
and  place  its  root  on  the  right,  after  the  manner  of  a  quotient 


9b,  Will  the  reasoning  in  the  example  apply  to  more  than  two 
peritnlfl  f 

8 


1(52  ELEMENTARY     ALWEBKA. 

in  division.  Subtract  the  square  of  (his  root  from  the  frat 
period^  and  to  the  remainder  bring  down  (he  second  period  for 
a  dividend. 

III.  Double  the  root  already  found^  and  place  it  on  the  left 
for  a  divisor.  Seek  how  many  times  the  divisor  is  contained 
in  the  dividend,  exclusive  of  the  right-hand  figure^  and  place 
the  figure  in  the  root  and  also  at  the  right  of  the  divisor. 

IV.  Multiply  the  divisor^  thus  augmented^  by  the  last  figure 
of  the  root^  and  subtract  the  product  from  the  dividend^  and  to 
the  remainder  bring  down  the  next  period  for  a  new  dividend. 
But  if  any  of  the  products  should  be  greater  than  the  divi- 
dend^ diminish  the  last  figure  of  the  root  by  one. 

V.  Double  the  whole  root  already  found^  for  a  new  divisor 
and  continue  the  02)eration  as  before^  until  all  the  2}^riods  an 
brought  down. 

97.  1st.  Remark. — If,  after  all  the  periods  are  brought 
down,  there  is  no  remainder,  the  proposed  number  is  a  per- 
fect square.  But  if  there  is  a  remainder,  you  have  only 
found  the  root  of  the  greatest  perfect  square  contained  in 
the  given  number,  or  the  entire  part  of  the  root  sought. 

For  example,  if  it  were  required  to  extract  the  square 
root  of  6G5,  we  should  find  25  for  the  entire  part  of  the 
root,  and  a  remainder  of  40,  which  shows  that  665  is  not 
a  perfect  square.  But  is  the  square  of  25  the  greatest  per- 
fect square  contained  in  665  ?  that  is,  is  25  the  entire  part 
of  the  roof?  To  prove  this,  we  will  first  show  iha-t,  the 
difference  between  the  squares  of  two  consecutive  numbers^  it 
equal  to  twice  the  less  number  augmente  I  by  one. 

96t  Give  the  rule  for  extracting  tbe  square  root  of  numbers.  What  \b 
die  first  step  ?  What  the  second  ?  What  the  third  ?  What  the  fourtl  ; 
What  the  fifth » 


BXTRACTION  OF  THE  SQUARE  KOOT.    1G3 

Let  .  .  a  z=     the  less  number, 

and  .  .  a  -|-  1     =     the  greater. 

Then  .  («  +  1)2  =  a^  +  2a +  1, 

and  .  .         (aY  =  a\ 

Their  difTerence  =       2a  -f  1     as  enunciated. 

Hence,  the  entire  part  of  the  root  cannot  be  augmented 
unless  the  remainder  is  equal  to  or  greater  than  twice  the 
root  found,  plus  one. 

But  25  X  2  +  1  =  51  >  40  the  remainder:  therefore, 25 
is  the  entire  part  of  the  root. 

98.  2d  Remark. — The  number  of  places  of  figures  in  the 
root  will  always  be  equal  to  the  number  of  periods  Into 
which  the  given  number  is  separated. 


EXAMPLES. 

1.  To  find  the  square  root  of  7225.  Ans.  85 

2.  To  find  the  square  root  of  17G89.  Ans.  133. 

3.  To  find  the  square  root  of  994009.  Ans.  997, 

4.  To  find  the  square  root  of  85G7353G.  Ans.  9250. 

5.  To  find  the  square  root  of  G7798756.  Ans.  8234. 

6.  To  find  the  square  root  of  978121.  A7is.  989. 

7.  T(.  find  the  square  root  of  95G484.  Ans.  978. 

8.  What  is  the  square  root  of  8G3729G1  1  Ans.  G031. 

9.  What  is  the  squarQ  root  of  22071204  ?  Ans.  4G98. 

10.  What  is  the  square  root  of  10G929  ?  Ans.  327. 

11.  What  is  the  square  root  of  120888G8379025  ? 

A?is.  347G905 


08.  Uow  laauj  %uro8  will  you  always  fiud  in  tbe  root  I 


164 


ELEMENTARY     ALGEBRA. 


99.  3d  Remark. — If  the  given  number  has  not  an  exact 
root,  there  will  be  a  remainder  after  all  the  periods  are 
brought  down,  in  which  case  ciphers  may  be  annexed,  form- 
ing new  periods,  for  each  of  which  there  will  be  one  deci- 
Tial  place  in  the  root. 

1.  What  is  the  square  root  of  36729  1 


In  this  example  there  are 
f^'o  periods  of  decimals, 
and  hence,  two  places  of 
decimals  in  the  root. 


3  67  29 

1 


191.64+ 


2  9'2Q7 
1261 

38  1  629 
381 

382  6 


24800 
22956 


3832  4 


184400 
15329^ 
31104  Rem. 


2.  What  is  the  square  root  of  2268741  ? 

3  What  is  the  square  root  of  7596796  ? 

4.  What  is  the  square  root  of  96  ? 

5.  What  is  the  square  root  of  1*53  ? 
6*  What  is  the  square  root  of  101  ? 


Ans.   1506.23+. 

Ans.  2756.22  4 . 

Ans.  9.79795  +. 
Ans.  12.36931  +. 
A71S.  10.04987  +. 


yy.  How  will  you  find  the  decimal  part  of  the  root  I 


EXTRACTION     OP     THE     SQUARE     ROOT         163 

7.  What  is  the  square  root  of  28507039G644 1 

Ans.  5347G2. 

8.  What  is  the  square  root  of  41005800G25  1 

Ans.  203975. 

9.  What  is  the  square  root  of  4830358420G084  ? 

Ans.  695007a 

Extraction  of  the  square  root  of  Fractions. 

100.  Since  the  square  or  second  power  of  a  fraction  is 
obtained  by  squaring  the  numerator  and  denominator  sepa- 
rately, it  follows  that  the  square  root  of  a  ftrction  will  be 
equal  to  the  square  root  of  the  numerator  divided  by  the 
square  root  of  the  denominator. 

For  example,  the  square  root  of    —    is  equal  to   — :  fcf 
a         a         o?- 

1.  What  is  the  square  root  of    — -? 

4 

2.  What  is  the  square  root  of    —  1 

64 

3.  What  is  the  square  root  of    — —  % 

81 

256 

4.  What  is  the  square  root  of    ^tt-^ 

ool 

1  /» 

5.  What  is  the  square  root  of     — -  ? 

64 


lOOi  If  the   numerator   and   denominator  of  a  fraction  arc    fM-riec 
Bquarcj*,  how  will  you  extract  the  square  rxxit  f 


Ans. 

2 

Ans. 

3 
4 

Ans. 

8 
9 

Ans. 

le 

18 

Ans. 

1 

2 

160  ELEMENTARY     ALGEBRA. 

«     ^Tx,         .       ,  r-       4096    „  ,  64 

6.  vV  hat  IS  the  square  root  oi      ■  i  Ans.  -— . 

7.  What  IS  the  square  root  01     ^^  ,^.^^  ?  Ans.  --— • 

^  950484  978 

101.  If  neither  the  numerator  nor  the  denominator  is  a 
perfect  square,  the  root  of  the  fraction  cannoi  be  exactly 
found.  We  can,  however,  easily  find  the  approximate  rout. 
For  this  purpose, 

MultipJi/  both  terms  of  the  fraction  hy  the  denominator^ 
which  makes  the  denominator  a  pterfect  square  without  altering 
the  value  of  the  fraction.  T/>en,  extract  the  square  roc*  of  the 
numerator^  and  divide  this  root  hij  the  root  of  the  denomina- 
tor ;  this  quotient  will  he  the  ajyproximate  root. 

Thus,  if  it  be  required  to  extract  the  square  root  of  — » 

15 

we  multiply  both  terms  by  5,  which  gives     —  • 

We  then  have 

-/Is  '^  3.8729  +  : 

hence,  3.8729  +  H-  5  =  .7745   +  =  Ans, 

7 

2.  What  is  the  square  root  of    —  %         Ans.  1.32287  -f . 

14 

3.  What  is  the  square  root  of    —  ?       ^ns.  1.24721  +. 


4.   What  is  the  square  root  of    Ht^? 


16 

Ans.  3.41869  -f-. 


101  •  If  the  numerator  and  denominator  of  a  fraction  are  not  perfect 
•jquare^,  how  do  you  extract  the  square  root  ? 


EXTRACTION  OF  TUK  SQUARE  ROOT.    aC«7 

13 

5.  What  is  the  square  root  of  7—  ]         A71S.  2.71313  4-. 

15 

6.  What  is  the  square  root  of   8—?         Ans.  2.88203  -f - 

5 

7.  What  is  the  square  root  of    —  ?         Ajis.  0.64549  -H 

8.  What  is  the  square  root  of    10—- 1 

Ans.  3.20936  +. 

102.  Finally,  instead  of  the  last  method,  we  may,  if  we 
pi<?ase, 

Chaiif/e  the  vrdffar  fraction  into  a  decimal^  n/id  continue  the 
Jiv7>ticn  until  the  nuinber  of  decimal  places  is  double  the  nmn- 
her  <]f  places  required  in  the  root.  Then,  extract  tJie  root  of 
the  dr^iinal  by  the  last  rule. 

Ex.  1.  Extract  the  square  root  of    — -     to  within   .001. 

14 

This  number,  reduced  to  decimals,  is  0.785714  to  withfn 

0.000001  ;  but  the  root  of  0.785714  to  the  nearest  unit,  is 

«86;  hence  0.886  is  the  root  of    —    to  within  .001. 

14 

2.  Find  the  \/2—    to  within   0.0001. 

Ans.   1.6931  -f . 

3.  What  is  the  square  root  of    —  ?        Ans.  0.24253  -f-. 

7 

4.  What  is  the  square  root  of  — -  ?         Ans.  0.93541  -}-. 

0 

5 

5.  What  is  the  square  root  of  —  I         Ans.  1.29099  +, 

o 


102.  By  what  other  method  may  the  root  be  ffjtuid  t 


1C8  ELEMENTARY     ALGEBRA. 


Extraction  of  the  Square  Root  of  Monomials. 

103.  In  order  to  discover  the  process  for  extracting  the 
square  root,  we  must  see  how  the  square  of  a  monomial  is 
formed. 

By  the  rule  for  the  multiplication  of  monomials  (Art.  35), 
we  have 

{paWcy  =  5a263c  x  5a^h  =  250*^^6^2  . 

that  is,  in  order  to  square  a  monomial,  it  is  necessary  to 
square  its  co-efficient^  and  double  the  exponents  of  each  of  tJi4 
different  letters.  Hence,  to  find  the  square  root  of  a  mono- 
mial, we  have  the  following 

RULE. 

I.  Extract  the  square  root  of  the  co-efficient, 
II.  Divide  the  exponent  of  each  letter  by  2. 


Thus,      -v/G4^*  =  ^aW    for     ^aW  x  ^aW  =  64a«6*. 

2.  Find  the  square  root  of  625a26V.  Ans.  25aMc^. 

3.  Find  the  square  root  of  hl^a'^h^c^.  Ans.  24a'^b^c^. 

4.  Find  the  square  root  of   i9Gx^i/^z'*^.  Ans.  14:X^yz'^, 

5.  Find  the  square  root  of  44:la.%^c^^d'^^. 

Ans.  21a*6V</8. 

6.  Find  the  square  root  of  IS^a^^b^'^c^^d'^ 

Ans.  2Sa%'^c^d. 

7.  Find  the  square  root  of  SI a%^c^, 

Ans.  9a'6V. 


103.  How  do  you  extract  the  square  root  of  a  monomial 


EXTRACTION  OK  THE  SQUARE  ROOT     169 

104.  From  the  preceding  rule  it  follows,  that  vhen  a 
monomial  is  a  perfect  square,  its  numerical  co-efficient  is  a 
verfcct  square  and  all  its  exponents  even  numbers.  Thus, 
■I'id^b'^  is  a  perfect  square ;  but  DSai*  is  not  a  perfect  square, 
because  V8  is  not  a  perfect  square,  and  a  is  afiected  with 
an  uneven  exponent. 

In  the  latter  ciise,  the  quantity  is  introduced  into  the  cal- 
culus by  affecting  it  with  the  sign  -y/  ,  and  it  is  written 
thus : 

Quantities  of  this  kind  are  called  radical  quantities^  or  irra 
tional  qzianiities,  or  simply  radicals  of  the  second  decree. 
They  are  also,  sometimes  called  Surds. 

Such  expressions  may  often  be  simplified,  by  employing 
the  principle  that,  the  square  root  of  the  product  of  two  or 
more  factors  is  equal  to  the  product  of  the  square  roo*"  of 
these  factors;  or,  in  algebraic  language, 


'^ahcd  .  .  .    zn^a.^/h.^/c.  ^d 


This  being  the  case,  the  above  expression,    y98a6*      'an 
be  put  under  the  form 

V41)6*  X  2a  =  V495*  x  -/2^ 

Now,  -/496*,  may  be  reduced  to  76^  j  hence. 


-/lJ8aF=762'v/2a. 
In  like  manner, 

-y/^baWcH  =  -v/Oo^iV  X  5bd  =  Sabc  ^553. 

^iili^i^Z^=  y/TUa^b*c^''  X  (Jbc  =  lUa b~c^  ^U. 
8 


170  ELEMENTARY     ALCJEBRA. 

The  quantity  which  stands  without  the  radical  sign  h 
called  the  co-ejicient  of  the  radical.  Thus,  in  the  expres- 
sions 

the  quantities  7*^2,  oabc,  I2ab\^^  are  called  co-efficients  of  tke 
radicals. 

Hence,  to  simplify  a  radical  expression  of  the  second 
degree,  we  have  the  following 

RULE. 

I.  Separate  the  expression  under  the  radical  sign  into  two 
factors^  one  of  which  shall  he  a  perfect  square. 

II.  Extract  the  square  root  of  the  perfect  square^  arid  then 
multiple/  the  root  by  the  indicated  square  root  of  the  remaining 
factors. 

105.  Remark. — To  determine  if  a  given  number  has  any 
factor  which  is  a  perfect  square,  we  examine  and  see  if  it  is 
divisible  by  either  of  the  perfect  squares 

4,     9,     16,     25,     36,     49,     64,     81,  &c., 

and  if  it  is  not,  we  conclude  that  it  does  not  contain  a  factor 
which  is  a  perfect  bquare. 


104.  When  is  a  monomial  a  perfect  square  ?  When  it  is  not  a  perfect 
square,  how  is  it  introduced  into  the  calculus  ?  What  are  quantities  of 
this  kind  called  ?  May  they  be  simplified  ?  Upon  what  principle  I 
Wh-at  IS  a  co-efficient  of  a  radical  ?  Give  the  rule  for  reducing  radi- 
nals. 

105.  How  do  you  determine  whether  a  given  number  has  a  factoi 
wh'-ch  is  a  perfect  square  ? 


KXTRACTION     OV     THE     SQUARE     ROOT.        171 


EXAMPLES. 


1.  Reduce     yTSo^    to  its  simplest  form. 

Arts.  5a  y/SoJc 

2,  Reduce     -y/T286*a^    to  its  simplest  form. 

Ans,W^a?d^, 


3.  Reduce     ^o'Za^h^c     to  its  simplest  form. 


4.  Reduce     y^250tt^6V     to  its  simplest  form. 

Ans.  IGaftV. 


5.  Reduce      ■y/1024a^6V     to  Its  simplest  form. 

Ans.  'S'2a<bh'^^/abe. 

6.  Reduce     ^l)l^da' b^cH     to  its  simplest  form. 

Ans.  270^6 Vy'aiZ 

7.  Reduce     -y/G75a"6V(/    to  its  simplest  form. 

Ans.  X^a^b'^c  y/Zcin. 

8.  Reduce     -y/1445aVc/*     to  its  simplest  form. 

-4n5.  17acV*A/5a 


9.  Reduce     yTOOSo^tTm®     to  its  simplest  form. 

Ans.  l2a*cPm*  y/lal, 

10.  Reduce     y^215Ga'*'i''c^     to  its  simplest  form. 

Ans.  14a'^6Vv/rr. 

11.  Reduce     -^iOba'b^d^     to  its  simplest  form. 

Ans.  ^a^b^d^y^^ 


1 72  ELEMENTARY     ALGVBRA. 

106.  Since  like  signs  in  two  facto-s  give  a  plus  sign  in 
the  product,  the  square  of  —  a,  as  well  ss  that  of  -|-  a,  wiL' 
be  a^\  hence,  the  square  root  ol  a?-  is  -^-'ther  -f  a  or  —  a 
Also,  the  square  root  of  '^ba^b^  is  either  -|-  ^^aU^  or  —  bah"^ 
Whence  we  may  conclude,  that  if  a  monoio'il  is  positive, 
its  square  root  may  be  affected  either  with  th*  sign  -f-  or  — 
thus,  Y^9a*  =  zL  3a^  ;  for,  +  oa^  or  —  3a^,  squared,  g.vet' 
9a*.  The  double  sign  rfc:  with  which  the  rout  \  affected,  it 
read  jo/w.s-  or  minus. 

If  the  proposed  monomial  were  negative^  it  w  ?Mld  be  im- 
possible to  extract  its  square  root,  since  it  hat  j  \«^t  beeo 
shown  that  the  square  of  every  quantity,  whethci"  jtVtivtf 
or  negative,  is  essentially  positive.     Therefore, 


are  algebraic  symbols  which  indicate  operations  that  cannof 
be  performed.  They  are  called  imaginary/  quantities,  or 
rather,  imaginary  expressions,  and  are  frequently  met  with 
in  the  resolution  of  equations  of  the  second  degree.  Thc^e 
symbols  can,  however,  by  extending  the  rules,  be  simplified 
in  the  same  manner  as  those  irrational  expressions  which 
indicate  operations  that  cannot  be  exactly  performed.  Thus, 
-y/—  9  may  be  reduced  by  (Art.  104).     Thus, 

and       -/  —  4a2  =  -/io^  x  -/  —  1  =  2a  -y/  —  \  :        also, 


-yf—^a^b  =  -v^4a2  X  -  26  =  2a  -/=^=  2a  y/2b  X  -v/^- 


106.  Wliat  sign  is  placed  before  the  square  root  of  a  monomial 
Why  may  you  place  the  sign  plus  or  minu8  ?     What  is  an  imagiuan 
quariLity  ?     Why  is  it  called  imaginary  ? 


RADICALS     OF     THE     SECOND     DEGREE.         173 


Of  Hie  Calculus  of  Radicals  of  the  Second  Degiee. 

107.  A  radical  quantity  is  the  indicated  root  of  an  im 
perfect  power. 

The  extraction  of  the  square  root  gives  rise  to  such  expres- 

sions  as  -y/oj  3y^  '^y^  which  are  called  irrational 
qiutntitien^  or  radicals  of  the  second  degree.  We  will  now 
estal)lish  rules  for  performing  the  four  fundamental  opera- 
tions of  Algebra  upon  such  expressions. 

108.  Two  radicals  of  the  second  degree  are  similar^  when 
the  quantities  under  the  radical  sign  are  alike  in  both.    Thus, 

3y^  and   5c yT  are   similar   radicals;   and   so   also  are 

Qy^  and   7^/27 

Addition. 

109.  Radicals  of  the  second  degree  may  be  added  togethei 
by  the  following 

RULE. 

I.  If  the  radicals  are  similar  add  their  co-efficients^  and  to 
1f\e  Slim  annex  the  common  radical. 

II.  If  the  radicals  are  not  similar^  connect  them  togethet 
Vfith  their  proper  signs. 

Thus,  3a  y^+  5c  ^=  (3a  +  5c)  -/ST 


107    What  is  a  radical  quantity  ?     What  are  such  quantities  called  I 
108.  When  are  radicals  of  the  second  degree  similar  t 
109i  How  do  you  ad</  similar  radicals  of  the  second  degree  ?     JIO10 
do  yuu  add  radicals  which  are  not  similar  t 


174  ELEMENTARY     ALGEBRA, 

IiJ  like  manner, 

7  V^+  3  v^  =(7  4-3)  V^  =  10  V^. 

Two  radicals,  which  do  not  appear  to  be  similar  at  first 
Bight,  may  become  so  by  simplification  (Art   104). 

For  example, 
V'48a62  +  6  V75a  =  46  V^-f  5&V^  =  9iV3al 
and         2  ^/^b  +  3  ^/b  =  (y^+Z^=9-y/o'. 

When  the  radicals  are  not  similar,  the  addition  or  sub- 
ti  iction  can  only  be  indicated.  Thus,  in  order  to  add 
3  \/b~  to    5  "v/a,    we  write 

5  Va  +  3  ^b7 

EXAMPLES. 

1.  What  is  the  sum  of     ^27a^    and     y'iSo^"'? 

Ans.  7a\/T. 

2.  What  is  the  sum  of     ^/EOa^    and     y/72a*b^  ? 

Ans.  lla^Sy'Sr 


/3a2  /  o2 

3. What  is  the  sum  of  Y  ""5 —  ^^^    \  "Tg" 


Ans.  4.a 


4.  What  is  the  sum  of    .y/l25     and     v^OOoZ'? 

-4nf.  (5  +  10a)  y^S 


RADICALS  OF  THE  SECOND  DEGRKB.    175 

5.  What  is  the  sum  of  .  /—    and   .  /  —  } 
V    147  V  2i>4 


10 


6.  What  is  the  sum  of    -/98a2j;    and  y'SOi:^  _  seJzj 

^«5.  7a-/2r"H-  G-Zur'-^- a*. 


7.  What  is  the  sum  of    yOSa^j:    and  ySSSo*^*  ? 

^/w.  (7a  +  12aV-)v/2l 

8.  Require!  the  sum  of  -y/72    ar  d  -y/128. 

^n5.   14  v^.' 

9.  Required  the  sum  of    -y/27     and   y^l47. 

^««.   lOv/sT 

10.  Required  the  sum  of    \/ t  ^"^  \/ • 

11.  Required  the  sum  of    2^/0^^    and     3-/646x*. 

^n^.  (2a +  24x2)^/67 

12.  Required  the  sum  of    '/243     and     10-/o03. 

^n5.  119  V^. 

3.  What  is  the  sum  of    v^320^2     ^nd  y/^Abc^l 

Ans.  (8a6  +  7a*P)v^ 


.4.  What  is  the  sum  of    /75^    and    v^SOOa^i*  ? 

^W5.  (5a^6J  4-  10a352).^/3S 


176  ELEMENTARY     ALGEBRA, 

Syhtraction. 

110.  To  subtract  one  radical  expression  from  another 
we  have  the  following 

RULE. 

I.  If  the  radicals  are  similar^  subtract  their  co-ejicients,  and 
to  the  difference  annex  the  common  radical. 

II.  If  the  radicals  are  not  similar,  indicate  their  differenct 
by  the  minus  sign, 

EXAMPLES. 

1.  What  is  the  difference  between    3a -^/T  and    a^^~b\ 
Here,     3a  -y/T  —  a  ^/~b  =  2a  -^fT    Ans, 

2.  From    9a -y/SW   subtract   6a  y^W. 

First,     9av^2762=:27a6-/37and   Ga-^/^^^  =  lSaby'Y; 
and  27a6yT—  ISaJy^  =  9a6-/^  Ans. 

3.  What  is  the  difference  of  -/TS"  and  y^Sl 

Ans.  y^sT 


4.  What  is  the  difference  of  y24a^  and    ^54b*  ? 

Ans.  (2a6-362)y^ 


110.   How  do  you  subtract  similar  radicals?     How  do  ycu  subtiac» 
radicals  which  arc  not  eimilar  ? 


RADICALS  OP  TUB  BECOKD  DEGREE.    177 


5.  Required  the  difference  of  \/  -^    and    kJ ^ — 

Ans,  —  v^*  * 
45  V  1. 


C    What  is  the  difference  of     vT282^    and     v'32^ 

Ans.  {%ab  -  4a*)  ^/2a 
7    What  is  the  difference  of    -/48a^P    and     -y/^ab  1 

Ans.  4rt6-y/3(/6  —  S\/^ 

8.  What  is  the  difference  of   -v/242"a*6*    and     -^a-'b^ 

Ans.  (\\a^b^-ab)^2^ 

9.  What  is  the  difference  of  \/ —     and  \/— "? 

V   4  V   9 

A?is V^ 

6 

10.  What  is  the  difference  of  -/320a2    ^nd    -/SOa^'i 

Ans.  4a-y/"5, 

11.  What  is  the  difference  between 

■y/TSO^P     and     -/245a6c2rf2  ? 

^/is.  (12a6  -  led)  -x/bah, 

12  What  is  the  difference  between 

yl)08a262     and     V'20flCT2  t 

yln«.  12a6v/2. 

13  What  is  the  diff*erence  between 


yTl2a»F     and     y^SoSp  1 
8» 


178  ELEMENTARY   ALGEBRA. 


Multiplication. 

111.    For  the   multiplication  of  radicals,  we  have   the 
following 

RULE. 

I.  Multiply  the  quantities  under  the  radical  signs  together^ 
and  place  the  common  radical  sign  over  the  product. 

II.  If  the  radicals  have  co-efficients^  we  multiply  them  to- 
gether^ and  place  the  product  before  the  radical  part. 

Thus,  yVx  V^=z  -v/^; 

This  is   the  principle  of  Art.   104,  taken  in  the  inverse 
order. 

EXAMPLES. 


1.  What  is  the  product  of    3   ySai    and    4Y/20a'? 

Ans.  120a  -/X 

2-  What  is  the  product  of   2rt  '\/hc     and    3a  -y/hc  ] 

Ans.   Qa^bc, 


3.  What  is  the  product  of  ^a^a^+b^  and   — 3ay^a2-h  h'^l, 

Ans.   —  6a2  {a^  -f  h^). 


111.  How  do  you  multiply  quantities  which  are  under  radical  siga*  ? 
When  the  radicals  ha^e  co-efficients,  how  do  you  multiply  tliem 


RADICALS  OF  THI  8H00ND  DEORKK.   179 

4.  What  is  the  product  of  3  ^/T  and  2-/8  ? 

Am,  24. 

5.  What  is  the  product  of  \^/^^   and    ^^^/\^  % 

Arts,  ^^abc-^lb, 

6.  What  is  the  product  of  2ar  +  -/i   and    2;r  —  -v/TT  ? 

Alls.  4x2  _  I 

7.  What  is  the  product  of 


Va4^2vT    and     Vo  — 2v/6  ? 

Ans.  -y/u?  —  4^. 

8.  What  is  the  product  of   3a-/27a3    by     -/2^1 

Division. 

112.  To  divide  one  radical  by  another,  we  have  the  fol- 
lowing 

RULE. 

I.  Divide  one  of  the  quantities  under  the  radical  sign  by 
the  other  J  and  place  the  common  radical  sign  over  the  quotient, 

II.  If  the  radicals  have  co-efficients^  divide  the  co-efficient  of 
the  dividend  by  the  co-efficient  of  the  divisor,  and  place  the 
quotient  before  the  radical,  found  as  above. 


112    How  do  you  divide  quantities  which  are  under  the  radical  sign ! 
When  tl»e  radicals  have  co-efficients,  how  do  you  divide  tliem  I 


180  ELEMENTARY     ALGEBRA, 


Thus,     ^       —\    — :     for   the   squares   of   these    two 
^/h       V   h 

expressions   are    each    equal    to    the    same  quantity    —j 

hence  the  expressions  themselves  must  be  equal. 


EXAMPLES. 


1.  Divide     5a  yo     by     26  ■\/c.  Am.  oTV 


5a    /  6 
c 


2.  Divide     12ac  ^J~m     by     4c  y^         ^n5.  8a  -/Sc. 
S.Divide     6a -/OOF    ly     3  y^Sft^;  ^r^s.  4a6  yT 


4.  Divide     4a2y^50P"    by     2a2y^.        Ans.  2^>2yTa 


5.  Divide     l^a^-y/^W^    by     13a-/9a6. 


6.  Divide     84a3J*-y/27ac     by     42aftx/32; 

7.  Divide     V^    by     y^  Ans.  \a. 


8.  Divide     Ga262y'20a3     by     12-v/5a.  ^?is.  aS^a 

9.  Divide     Qa^/Jm     by     S^/sT  ^;is.  2a^»-/2 

10.  Divide     485*-v/T5     by     ^b'^^/^,  Ans.  3606? 

11.  Divide     ^a%H--/ld^    by     2a'/28^ 

^W5.  2a6*c3(i 

12.  Divide     ma^c^^/^m     by     4S*f^>'C-/26. 

Ans.  Ha^^r' 


RADICALS 


UF  THE  SECOND  DEGREE.    181 


13.  Divide     27«'''6«-/21a3     by     ^/la. 

Am.  9^6^ -/a] 
U.  Divide     \%a%^^/~^    by     Q,ah^/^. 

To  ExtrcLct  the  Square  Boot  of  a  Polynomial. 

113.  Before  explaining  the  rule  for  the  extraction  of 
thd  square  root  of  a  polynomial,  let  us  first  examine  the 
•quares  of  several  polynomials  :  we  have 

(a  +  hf  =  «2  +  2ab  +  h\ 

(a  +  i  +  c)2  =  a2  +  2ah  -^  b"^  +  2{a  -f-  h)c  -f-  c^, 

{a-{-b-\-c  +  df  =  a'^-{-'iab-^h'^-\-  2(a  -f-  d)c  -h  c» 

+  2(a  +  &  4-  c)d  +  d^. 

The  law  by  which  these  squares  are  formed  can  be  enun- 
ciated thus : 

The  squure  of  any  polynomial  is  equal  to  the  square  of  the 
first  ternu,plus  twice  the  jjroduct  of  the  first  term  by  the  second^ 
plus  the  square  of  the  second ;  plus  twice  the  first  tivo  terms 
iiiultiplied  by  the  third^plus  the  square  of  the  third ;  plus  twice 
the  first  three  terms  multiplied  by  the  fourth^  plus  the  square 
of  the  fourth  ;  atid  so  on. 


113.  WLat  is  the  square  of  a  linomial  equal  to  What  is  the 
square  of  a  trinomial  equal  to  I  What  ia  tlie  square  of  any  polynomial 
equal  to  f 


182  ELEMENTARY     ALGEBRA, 

114.  Hence,  to  extract  the  square  root  of  a  po.ynomial, 
we  have  the  following 

KULE. 

I.  Arrange  the  polynomial  ivith  reference  to  one  of  its  let- 
ters^ and  extract  the  square  root  of  the  first  term :  this  will 
give  the  first  term  of  the  root. 

II.  Divide  the  second  term  of  the  polynomial  by  doable  the 
first  term  of  the  root,  and  the  quotient  will  be  the  second  term 
of  the  root. 

III.  Then  form  the  square  of  the  sum  of  the  two  terms  of 
the  root  found,  and  subtract  it  from  the  first  polynomial,  and 
then  divide  the  first  term  of  the  remainder  by  double  the  first 
term  of  the  root,  and  the  quotient  will  be  the  third  term. 

IV.  Form  the  double  product  of  the  sum  of  the  first  and 
second  terms  by  the  third,  and  add  the  square  of  the  third ; 
then  subtract  this  result  from  the  last  remainder,  and  divide 
the  first  term  of  the  result  so  obtained  by  double  the  first  term 
of  the  root,  and  the  quotient  will  be  the  fourth  term.  Then 
proceed  in  a  similar  manner  to  find  the  other  terms. 

EXAMPLES. 

1.  Extract  the  square  root  of  the  polynomial 

AQaW  —  24ab^  +  25a*  -  SOa^  +  165*. 
First  arrange  it  with  reference  to  the  letter  a. 

5a2  _  Sab  4  4.V 


25a*  —  SOa^  +  49aW  —  24a63  —  166* 
25a*  — 30a36+    9a^^^ 


10a2 


40aW  -  24ab^  +  166*    1st  Rem. 
40aW  -  24a63  +  166* 

(r~.     '.     7    2d  Rem. 


»      IIADICALS     OF    THE    BKCOND     DEGREE.  185 

After  having  arranged  the  polynomial  with  reference  to 
a,  extract  the  square  root  of  25a*;  this  gives  fja^,  which  is 
placed  at  the  right  of  the  polynomial :  then  divide  the  second 
term,  —  *SQa\  by  the  double  of  Sa^,  or  lOa^  ;  the  quo- 
tient is  —  3a6,  which  is  placed  at  the  right  of  Sa^.  Hence, 
the  first  two  terms  of  the  root  are  Sa^  -r-  Sab.  Squaring 
this  binomial,  it  becomes  25a*  —  SOa^b  -\-  9aH^^  which,  sub- 
tracted from  the  proposed  polynomial,  gives  a  remainder^ 
of  which  the  first  term  is  AOa'^b'^.  Dividing  this  first  term 
by  10a2,  (the  double  of  Sa^),  the  quotient  is  -(-  4.b^ ;  this 
is  the  third  term  of  the  root,  and  is  written  on  the  right  of 
the  first  two  terms.  By  forming  the  double  product  of 
5a2  —  Sab  by  46^,  squaring  46^,  and  taking  the  sum,  w< 
find  the  polynomial  A^Oa^b'^  —  24aP  -f-  106*,  which,  sub 
tracted  from  the  first  remainder,  gives  0.  Therefore, 
Sa^  —  3ab  -f  46^  is  the  required  root. 

2.  Find  the  square  root  of  a*  -h  4a3a:  +  Qa^x^  +  4ax^  +  x*, 

Ans.  a?-  4-  2«ar  -h  a?^ 

3.  Find  the  square  root  of  a*  —  \aH-\-  Oa^x^  —  4ax^  -f  x*^ 

Atis.   a?  —  ^ax  -|-  x^. 

4.  Find  the  square  root  of 

4jc«  +  12j:6  4-  5ari  —  2ar3  +  7^2  -  2x  +  1. 

An^,    2a:3  _f.  3^  _.  a;  ^_  1. 

5.  Find  the  square  root  of 

Oa*  —  12a36  +  28a2i2  _  lOai^  -f  106*. 

Ans,   3a2-2a6  +  462 


111.  GiTe  the  rule  for  extracting  the  pquare  root  of  a  polynomial 
What  is  the  first  step  ?  Wliat  the  second  ?  What  the  third  I  Whal 
the  fourth  t 


184  ELEMENTARY      ALGEBRA. 

6.  What  is  the  square  root  of 

x^  —  4:ax^  +  4a2a;2  —  4:X^  -|_  8aa;  +  4 1 

1.  What  is  the  square  root  of 

9aj2  —  12ic  +  0:cy  +  y2  _  4y  4-  4-? 

Ans.  Sx  -\-  y  —  % 

8.  What  is  the  square  root  of  y*  —  2y^x^  -\-2x^  +  2^^ 
f-  1  4-  a;*  ?  u4?i5.    y^  —  x^  —  1. 

9.  What  is  the  square  root  of   9a*6*  -  SOa^i^  +  25a252  1 

-4?is.  3a262  —  5a6. 

10.  Find  the  square  root  of 

25a*62  _  40a362c  +  76a262c2  -  48a62c3  +  Z^h\^  —  30a*5c 
+  24a36c2  —  36a26c3  +  9a*c2. 

Ans.    ha?-h  —  3a2c  —  4a6c  +  66<?^. 

115.  We  will  conclude  this  subject  with  the  following 
icmarks : 

1st.  A  binomial  can  never  be  a  perfect  square,  since  we 
know  that  the  square  of  the  most  simple  polynomial,  viz : 
a  binomial,  contains  three  distinct  parts,  which  cannot  ex- 
perience any  reduction  amongst  themselves.  Thus,  the 
expression  a^  -f-  ^^  is  not  a  perfect  square ;  it  wants  the 
term   =fc2a6  in  order  that  it  should  be  the  square  of  a  =b  6. 

2d.  In  order  that  a  trinomial,  when  arranged,  may  be  a 
perfect  square,  its  two  extreme  terms  must  be  squares,  and 
the  middle  term  must  be  the  double  product  of  the  square 
roots  of  the  two  others.  Therefore,  to  obtain  the  square 
root  of  a  trinomial  when  it  is  a  perfect  square  :  Extract  the 
roots  of  the  two  extreme  terms,  and  give  these  roots  the  same 
or  contrary  sl(/ns,  according  as  the  middle  term  is  positive  or 


0>.' 

RADICALS     OF     THE     M<;ey^D     DSGKES.  lS[f 

negative.     To  verify  it,  see  if  the  do^^h^jpfoduct  af^tke  two 

roots  w  the^same  as  the  iniddUjerm  of  the  trinomial.     Thus, 

9^6  _  4^a^i2i^  (Aa2^4   ig  a  perfect  square, 

siiice        ^  V^  —  ^\  f  nd  -^Ma^b"-  =  —  Sab\ 

and  also  2  x  Sa^  x  —  Sab^  =  —  4Sa*b^  =  the  middle  term. 

JJut,  4a2  -f  14a6  -\-  9b'^  is  not  a  perfect  square  :  for, 
d  though  4a2  and  -f  9^2  are  the  squares  of  2a  and  36, 
yet   '.i  X  2a  X  36   is  not  equal  to    14a6. 

3d.  In  the  series  of  operations  required  by  the  general 
rule,  when  the  first  term  of  one  of  the  remainders  is  not 
exacily  divisible  by  twice  the  first  term  of  the  root,  we  may 
conclude  that  the  proposed  polynomial  is  not  a  perfect 
square.  This  is  an  evident  consequence  of  the  course  of 
reasoning,  by  which  we  have  arrived  at  the  general  rule  for 
extracting  the  square  root. 

4th.  When  the  polynomial  is  not  a  perfect  square,  it  may 
Bometimes  be  simplified.  (See  Art.  104.) 

Take,  for  example,  the  expression  -y/a^6  -f-  4^a'^b^  +  4a6'\ 

The  quantity  under  the  radical  is  not  a  perfect  square ; 
but  it  can  be  put  under  the  form  ab  (a^  -f  4a6  -f-  46^.) 
Now,  the  factor  within  the  parenthesis  is  evidently  the 
square  of  a  -f-  26,  w  hence  we  may  conclude  that, 

Va36  -f  4a262  -}-  4a63  =  (a  -f  26)  ya6. 


2.  Reduce  'y/2a^b^-4alJ^2b^  to  its  simple  form. 

A71S.  (a  —  6)  V^A; 

115.  Can  a  binomial  ever  be  a  perfect  power  I    Why  not  ?    When  in 
a  trinumial  a  f>erfect  f^quare  ?     When,  in  extracting  the  square  root,  we 
find  Uiat  the  first  term  of  the  remainder  is  not  diviaible  by  twice  th« 
loot,  ia  the  polyiiomial  a  perfect  power  or  wit  I 
0 


ISii  ELEMENTARY    ALUEBBA. 


?, 


CHAPTER  VI. 

Equations  of  the  Second  Degree. 

116.  An  Equation  of  the  second  degree  is  one  m  -which 
the  greatest  exponent  of  the  unknown  quantity  is  equal  to  2 

If  the  equation  contains  two  unknown  quantities,  it  is  of 
the  second  degree  when  the  greatest  sura  of  the  exponents 
with  which  the  unknown  quantities  are  affected,  in  any 
term,  is  equal  to  2.     Thus, 

x^  =  a,    ax^  +  ^^  =  c,     and    xy  -\-  x  =i  c?^^ 

are  equations  of  the  second  degree, 

117.  Equations  of  the  second  degree,  involving  a  single 
unknown  quantity,  are  divided  into  two  classes : 

1st.  Equations  which  involve  only  the  square  of  the  un- 
known quantity  and  known  terms.  These  are  called  Incom 
plete  Equations, 

2d.  Equations  which  involve  the  first  and  second  powers 
of  the  unknown  quantity  and  known  terms.  These  ars 
called  Complete  Equations, 


il6i  What  is  an  equation  of  the  second  degree  ? 
117i  Into  how  many  classes  are  equations  of  the  second  degree  di- 
vided ?    What  is  an  incomplete  equation  ?  What  is  a  complete  equaticwi  ? 


BQU^TIONB     OF     THE     SECOND     DEGREE.       1B7 

Thus,  x2  -f  2x2  -  5  =  7  -^ 

and  5x2  _  3^2  _  4  _  ^i 

are  incomplete  equations :  and 

:Vr2  —  5x   -  3x2  +  a  =  6 
2x^^  —  8x2—    X  —c  =  d 

are  complete  equations. 

0/  Incomplete  Equations, 

118.  If  we  take  an  incomplete  equation  of  the  form 
14x2  _  8^2  =  40  _  2x2 
we  have,  by  collecting  the  terms  involving  x*, 
8x2  _  40^  or  x2  =  5. 
Again,  ii   we  have  the  equation 

ax2  +  6x2  +  rf=/, 
we  shall  have, 

(a  -f  i)x2  =.f—dy  and  x2  =       .,  =  w, 

by  substituting  m  for  the  known  terms  which  compose  the 
second  member.     Hence, 

Every  incomj^lete  equation  can  be  reduced  to  an  equation 
involving  two  tervis,  of  the  form 
x2  =  m, 

and  from  this  circumstance  the  incomplete  equations  are 

often  called  equations  involving  two  terms. 

From  which  we  have,  by  extracting  the  square  root  of 

both  members,  ,      , — 

X  =  ±  y  m. 

1 18.  To  what  furm  may  every  incomplete  equation  be  reduced  I    Whai 
are  incomplete  equations  often  called  t 


188  ELEMENTARY     ALGEBRA. 

1.  What  number  is  that  which  being  multiplied  by  itself 
the  product  will  be  144  ? 

Let  X  =  the  number :  then 

ar  X  «  =  a;2  =  144. 

It  is  plain  that  the  value  of  x  will  be  found  by  extracting 
the  square  root  of  both  members  of  the  equation  :  that  is 

^/W—  •v/lii' :  that  is,  x  =  12. 

2.  A  person  being  asked  how  much  money  he  had,  said 
if  the  number  of  dollars  be  squared  and  6  be  added,  the 
sum  will  be  42 :  how  much  had  he  ? 

Let  X  =  the  number  of  dollars. 

Then,  by  the  conditions 

a;2  +  6  =  42: 

hence,  a;2  =  42  —  6  =  36. 

and  x  =  Q. 

Ans,  %Q. 

3.  A  grocer  being  asked  how  much  sugar  he  had  soldtc  a 
person,  answered,  if  the  square  of  the  number  of  pounds  be 
multiplied  by  7,  the  product  will  be  1575.  How  many 
pounds  had  he  sold  1 

Denote  the  number  of  pounds  by  x. 

Then  by  ths  conditions  of  the  question 

7a;2  =  1575 : 

hence,  x"^  =  225 

and  X  =zl5. 

Am,  15. 


EQUATIONS  OF  THE  SECOND  DBORSB.   189 

4.  A  person  being  nsked  his  age  said,  if  from  the  square 
of  my  age  you  take  192,  the  remainder  will  be  the  square 
of  half  my  age :  what  was  his  age  1 

Denote  his  age  by  x. 

Then  by  the  conditions  of  the  question 


-192=  (1.)- 


and  by  clearing  the  fractions 

4^2  _  768  =  x^ ; 
hence,  4z^  —  x^  =  7G8 

and  Sx^  =  7G8 

x^  =  256 
X   =    IG. 


Am.  li^ 


5.  What  number  is  that  whose  eighth  part  multiplied  b) 
its  fifth  part  and  the  product  divided  by  4,  shall  give  a  quo- 
tient equal  to  40 1 

Let  X  =  the  number. 


By  the  conditions  of  the  question 

H-4-)*- 

=  40, 

"■>■».           ,«-« 

by  clearing  of  fractions, 

x^  =  6400 

X  =    80. 

Am.  SO. 


190  ELEMENTART     i.IGEBBA» 

119.  Hence,  tc  find  the  value  of  x  we  have  the  follow 
ing 

RULE. 

I.  Find  the  value  of  x^ ;  and  then  extract  the  square  root 
of  both  members  of  the  equation. 

6.  What  is  the  value  of  x  in  the  equation 

3a:2  +  8  =  5a;2_.10. 

By  transposition  2x'^  ■—  5x^  =  —  10  —  8, 

by  reducing       *  —  2x'^  =  —  18, 

by  dividing  by  2,  and  changing  the  signs 

a;2  =  9, 

by  extracting  the  square  root,  x  =  S, 

We  should,  however,  remark  that  the  square  root  of  9,  is 
either     +3    or    —  3.     For, 

+  3x+3=9     and     — 3x— 3  =  9. 

Hence,  when  we  have  the  equation 

a;2  =  9, 
we  have,  x  =  +S     and    ^  =  —  3. 

120.  A  root  of  an  equation  is  any  expression  which  being 
substituted  for  the  unknown  quantity,  will  satisfy  the  equa- 
tion, that  is,  render  the  two  members  equal  to  each  other. 
Th  IS,  in  the  equation 

X^=:9 

there  are  two  roots,   +  3  and  --  3 ;    for  either  of  these 
aumbers  being  substituted  for  x  will  satisfy  the  equation. 


BQUATIONB     OF     THE     SSOONO     DEGREE.       191 


7.  Again,  if  we  take  the  equation 
ire  shall  have 


x^  =  m, 


ar  =  -f-  -y/^)  and    x  =  —  -^In, 
For,  (4--/^)  2  =  7/1; 

and  (  —  -r/lti  )^  =  m; 

Hence,  we  may  conclude, 

1st.   That  every  inc(yni2)lete  equation  of  the  second  degree 
ha^  two  roots. 

2d.  That  these  roots  are  numerically/  eqiuzl^  but  have  con- 
irary  signs. 

8.  What  are  the  roots  of  the  equation 

3x2  -f  6  =  4x2  -  10. 

Ans.  x  =  +  4  and  a;  =  —  4. 

9.  What  are  the  roots  of  the  equation 

1  a:2 

_.._8  =  -+10. 

Ans.  ar  =  -f  9  and  a;  =  —  9. 

10.  What  are  the  roots  of  the  equation 

4x2  _|_  13  _  2x2  =  45. 

Ans.  X  =  4-  4  and  x  =  —  4, 

119.  How  do  you  resolve  an  incomplete  equation  t 

120.  What  is  the  root  of  an  equation?  What  are  the  roots  of  thp 
•iquation  x*  =  9  !  Of  the  equation  x*  =  m  f  How  many  roots  has  every 
'iacu:i>plete  eq-iatiuo  ?     How  do  those  roots  ecmpare  with  each  other  t 


192  ELEMENTARY     ALGEBRA. 

8.  What  are  the  roots  of  the  equation 

6a;2  -  7  =  Sx^  +  5. 

Ans.  X  =  +2,     X  =  -'-2 

9.  "What  are  the  roots  of  the  equation 

8  +  5a:2  =  ^  _|.  4^2  _|_  28. 
5 

Ans.  a;  =  +  5,     a;  =  —  5. 

10.  Find  a  number  such  that  one-third  of  it  multiplied 
by  one-fourth  shall  be  equal  to  108  ? 

Ans.  36. 

11.  What  number  is  that  whose  sixth  part  multiplie4  by 
its  fifth  part  and  product  divided  by  ten,  shall  give  a  quo- 
tient equal  to  3  1 

Ans.  30. 

12.  What  number  is  that  whose  square,  plus  18,  shall  be 
equal  to  half  its  square  plus  30  J 1 

Ans.  5. 

13.  What  numbers  are  those  which  are  to  each  other  as 
1  to  2  and  the  difference  of  whose  squares  is  equal  t    75  ? 

Let       X  =     the  less  number. 
Then  2x  =     the  greater. 
Then,  by  the  conditions  of  the  question 
4a:2  _  a;2  _  75^ 
hence,  Sx^  =  75 ; 

and  by  dividing  by   3,   x^  =  25    and   a?  =  5, 
and  2x=  10. 

Ans,  5  and  10 


EQUATIONS  OF  THE  SECOND  DEGREE.    193 

14.  What  two  numbers  are  those  which  are  to  each  other 
as  5  to  6,  and  the  difference  of  whose  squares  is  44  ] 

Let         X  =    the  greatest  number. 


Then,| 

X  =  the  less. 

By  the  conditions  of  the  problem 

by  clearmg 

of  fractions, 

36x2  _  25^2  =  1584  ; 

hence, 

1^2  =  1584, 

and 

x^  =  144, 

hence, 

x=l2, 

and 

i—- 

Ans.  10  and  12. 

15.  What   two   numbers   are   those  which   are  to  each 
other  as  3  to  4,  and  the  difference  of  whose  squares  is  28  ? 

Alls,  6  and  8 

16.  What  two  numbers  are  those  which  are  to  each  othei 
as  5  to  11,  and  the  sum  of  whose  squares  is  584  1 

A)is.  10  and  22, 

17.  A  says  to  B,  my  son's  age  is  one  quarter  of  yours, 

and   the   difference   between   the  squares  of  the  numbei*? 

ropresenting  their  ages  is  240  :  what  were  their  ages  ? 

.       {  Eldest       16 
Ajis.  < 

(  Younger     4 


194  ELEMENTARY     ALGEBRA, 


When  there  are  two  unknown  quantities, 

121.  When  there  are  two  or  more  unknown  quantities, 
eliminate  one  of  them  hy  the  rule  of  Article  77 :  there  tvill 
thus  arise  a  new  equation  with  hut  a  single  unknown  quantity^ 
the  value  of  which  may  be  found  by  the  rule  already  given, 

1  There  is  a  room  of  such  dimensions,  that  the  differ- 
ence of  the  sides  multiplied  by  the  less,  is  equal  to  36,  and 
the  product  of  the  sides  is  equal  to  360  ;  what  are  the 
sidjis? 


Let  X  =  the  less  side ; 

y  =  the  greater. 

Then,  by  the  first  condition, 

(y- 

x)x 

=  36; 

and  by  the  2d, 

xy 

=  360. 

From  the  first  equation. 

we  have 

xy  ■ 

-x' 

=  36; 

and  by  subtraction, 

x' 

=  324. 

Hence,                 x=  -^Z 

324 

=  18; 

y  = 

360 

18 

=  20. 

Ans.  a;  =  18,  y  =  20. 


121 1  How  do  you  resolye  the  equation  when  there  are  two  or  mora 
Kinknuwn  quantities  f 


BQUATI0N8  OF.THS  BECOND  DEORKK.   195 

2.  A  merchant  sells  two  pieces  of  muslin,  which  together 
measure  12  yards.      He    received  for  each  piece  just  ec 
many  dollars  per  yard  as  the  piece  contained  yards.     Now, " 
he  gets  four  times  as  much  for  one  piece  as  for  the  other : 
how  many  yards  in  each  piece  1 

Let    X  =    the  number  in  the  larger  piece ; 
y  =    the  number  in  the  shorter  piece. 
Then,  by  the  conditions  of  the  question, 
ar  4-  y  =  12. 
X  X  X  =  x^  =  what  he  got  for  the  larger  piece ; 
y  X  y  =:y^  =  what  he  got  for  the  shorter. 
ibid  x^  =  4y^,  by  the  2d  condition, 

X  =  2y,  by  extracting  the  square  root. 

Substituting  this  value  of  x  in  the  first  equation,  we  have 
y  +  2y  =  12 ; 
and  consequently,  y  =   4, 

and  x=   S, 

An8.  8  and  4. 

3.  What  two  numbers  are  those  whose  product  is  30,  and 
quotient  3 J  *?  Ans.  10  and  3. 

4.  The  product  of  two  numbers  is  a,  and  their  quotient 
h :  what  are  the  numbers  1 


Ans,   y/ah  and  \/ -r» 


5.  The  sum  of  the  squares  of  two  numbers  is  117.  and 
the  diilerence  of  their  squares  45  :  what  are  the  numbers  ? 

Ans.  9  and  G» 


196  ELEMENTARY     ALGEBRA. 

6.  The  sum  of  the  squares  of  two  numbers  is  a,  and  th^ 
diiference  of  their  squares  is  b  :  what  are  the  numbers  1 

.  /a~-l-  h  la  —  h 

Ans.  x=:  \/  — - — ,  y  z=  4  /  — — — 
V       2     '  ^      V       2 

7.  What  two  numbers  are  those  which  are  to  each  othei 
as  3  to  4,  and  the  sum  of  whose  squares  is  225  ? 

Ans.  9  and  12* 

8.  What  two  numbers  are  those  which  are  to  each  othei 

as  m  to  ?i,  and  the  sum  of  whose  squares  is  equal  to  a?  ? 

.  ma  na 

Arts.         ■  , 

y  m"^  -\-  -nP"       y  wi^  _j_  ^2 

9.  What  two  numbers  are  those  which  are  to  each  other 
as  1  to  2,  and  the  difference  of  whose  squares  is  75  % 

Ans,  5  and  10. 

10.  What  two  numbers  are  those  which  are  to  each  othe? 
as  m  to  71,  and  the  difference  of  whose  squares  is  equL^ 
to  62] 

.  mh  nh 

Ans.  v.^ 

y  m^  —  n^ '     y  m^  —  ti** 

11.  A  certain  sum  of  money  is  placed  at  interest  for  sr» 
months,  at  8  per  cent,  per  annum.  Now,  if  the  amount  bt 
multiplied  by  the  number  expressing  the  interest,  the  pro 
duct  will  be  $562500  :  what  is  the  amount  at  interest  ? 

Ans.  $3750 

12.  A  person  distributes  a  sum  of  money  between  a  num 
her  of  women  and  boys.  The  number  of  women  is  to  the 
number  of  boys  as  3  to  4.  Now,  the  boys  receive  one 
half  as  many  dollars  as  there  are  persons,  and  the  women 
twice  as  many  dollars  as  there  are  boys,  and  together  thej 
receive  138  dollars :  how  many  women  were  there,  and  ho\» 
many  boys? 

i3( 
(4^ 


J        ,  36  women 


48  boysk 


1QUATI0N8  OF  THE  SECOND  DEGREE.  101 

Of  Complete  Equations. 

122.  We  have  already  seen  (Art.  117),  Uiat  a  complete 
equation  of  the  second  degree,  contains  the  square  of  the 
unknown  quantity,  the  first  power  of  the  unknown  qunintity, 
and  known  terms. 

1.  If  we  have  the  complete  equation 

5x2  _  2x2  +  8  =  9j:  -f  32^ 
we  have,  by  transposing  and  reducing, 
3:c2  -  9x  =  24, 
and  by  dividing  by  3, 

5^»  —  3a:  =  8, 
an  equation  containing  but  three  terms. 

2.  If  we  have  the  equation 

a2a;2  -|-  Zahz  -\-  x^  z=:  ex  ■\-  d^ 
by  collecting  the  co- efficients  of  x^  and  ar,  we  hav ' 

(a?  -f-  1)2:2  ^  (3a5  _  c)x  =  rf; 

and  dividing  by  the  co-efficient  of  x^^  we  have 

-   .    Sab  —  c  d 

^   a2  -f  1  a2  4-  r 


122i  How  many  terms  does  a  complete  equation  of  the  second  tJegr*»e 
contaic  f  Of  what  is  the  first  term  compcsod  ?  Th«  second  I  T^xh 
tiur.lt 


198  KLEMBNTAHY      ALGEBRA. 

If  we  represent  the  co-efficient  of  x  by  2^,  and  the  known 
term  by  ^,  we  have 

x^  +  2pa;  =  2', 
an  equation  containing  but  three  terms :    hence, 

Every  complete  equation  of  the  second  degree  may  be  re- 
duced to  an  equation  containing  but  three  terms, 

123.  We  wish  now  to  show  that  there  maybe  four  forma 
under  which  this  equation  will  be  expressed,  each  depending 
on  the  signs  of  2p  and  q, 

1st.  Let  us,  for  the  sake  of  illustration,  make 
2p  =  4-  4,     and     g-  =  +  5 : 
wo  shall  then  have         a;^  +  4a;  =  5. 

2d.  Let  us  now  suppose 

2p  =  —  4,     and    g  =  -f  5 : 
we  shall  then  have        a;^  _  4^  _-  5^ 

3d.  If  we  make 

2p  =  4-  4,     and    gr  =  —  5, 
we  have  «2  -}-  4a;  =  -^  5. 

4th.  If  we  make 

2p  =  —  4,     and    g  ==  —  5, 
we  have  a;^  —  4x  =  —  5. 

123i  Under  how  many  forms  may  every  equation  of  the  second  de- 
gree be  expressed  ?  On  what  will  these  forms  depend  ?  What  are  the 
«igna  of  the  co-eflficient  of  x  and  the  known  term,  in  the  first  form! 
What  in  the  second  ?  Wha  in  the  third  ?  What  in  the  fourth  ?  Repeat 
the  four  ibrma 


EQUATIONS  OF  THE  SECOND  DEGREE.    109 

Wc  therefore  conclude,  that  every  complete  equation  of 
the  second  degree  may  be  reduced  to  one  of  these  forma : 

x^  +  ^px  =  -f  ^,  1st  form. 

«^  —  2/>2r  =  -f-  ^,  2d  form. 

x^  -f  ^jpx  =  —  y,  3d  form. 

x^  —  2px  =  —  g,  4th  form. 

124.  Remark. — If,  in  reducing  an  equation  to  either  of 
these  forms,  the  second  power  of  the  unknown  quantity 
should  have  a  negative  sign,  it  must  be  rendered  positive 
by  changing  the  sign  of  every  term  of  the  equation. 

125.  We  are  next  to  show  the  manner  in  which  the  value 
of  the  unknown  quantity  may  be  found.  We  have  seen 
(Art.  38),  that 

{x-\-pY  =  x^  +  2px-\-p^', 

and  comparing  this  square  with  the  first  and  third  forms,  we 
see  that  the  first  member  in  each  contains  two  terms  of  the 
square  of  a  binomial,  viz  :  the  square  of  the  first  term  plus 
twice  the  product  of  the  2d  term  by  the  first.  K^  then,  we 
take  half  the  co-efficient  of  a;,  viz  :  />,  and  square  it,  and  add 
(he  result  in  each  equation,  to  both  members,  we  have 

x^  -\r  2px  +  jp^  =  q  +  p^ 

X^  -\-  2pX  -}-  ^2  -_   _  g  ^p2^ 

\n  which  the  first  members  are  perfect  sqiares.     This  13 

124i  li  in  reducing  an  equation  to  either  of  these  forms  the  co-effi« 
dent  of  X*  is  negative,  what  do  you  do  f 

126>  What  is  the  square  of  a  binomial  equal  to?  What  does  the 
first  member  in  each  form  contain  7  How  do  you  render  the  first  mem' 
ber  a  perfect  square  ?     What  is  this  called  ? 


200  ELEMENTARY     ALGEBRA. 

called   completing   the   square.     Then,    by    extracting   the 
square  root  of  both  members  of  the  equation,  we  have 


X+pzizziz  y/g  +p\ 


and  X  -{-  p  =  dz  ■y-—g-\-p^i 

which  gives,  by  transposing  p, 


=  -2>-=^V9+P\ 


X  =z  —  p  ±  y  —  q  +  p"^- 

126.  If  we  compare  the  second  and  fourth  forms  with 
the  square 

(x  —  pY  =  x^  — 2px  +  p^, 

we  also  see  that  half  the  co-efficient  of  x  being  squared  and 
the  result  added  to  both  members,  will  make  the  first  mem- 
bers perfect  squares.     Having  made  the  additions,  we  have 

x^  — -  2px  -\-  p^  =  q  -{-  p^, 
x^  —  2px  -\-  p^  =.  —  q  -^p^. 

Then,  by  extractmg  the  square  root  of  both  members,  we 
have 


x—p=  ±V7  +  i?^ 


and  X  —p  =  ±:y^  q  -i-  p^'y 

and  by  transposing     —p,   we  find 


x=zpzh-^q-{-p^, 


and  X  z=p  ±  y  —  q+p^. 


126i  In  the  8ec<»id  form,  how  do  you  make  the  first  member  a  parked 
«quarei 


I- 


EQUATIONS  OF  THE  SECOND  DEOREB.   201 


127.  Hence,  for  the  resolution  of  every  equation  of  the 
second  degree,  we  have  the  following 


BULB. 


I.  Reduce  the  equation  to  one  of  the  four  forms, 

II.  Take  half  the  co-efficient  of  the  second  term^  square  it^ 
and  add  the  result  to  both  members  of  the  equation. 

III.  Then  extract  the  square  root  of  both  members  of  the 
equation  ;  after  which^  transpose  the  known  term  to  the  sccojid 
member. 

Remark. — The  square  root  of  the  first  member  is  alwayf 
equal  to  the  square  root  of  the  first  term,  plus  or  minus 
half  the  co-efticient  of  the  first  power  of  the  unknowD 
quantity. 

EXAMPLES    OF    THE    FIRST    FORM, 

1.  What  are  the  values  of  a;  in  the  equation 

2a:2  -I-  8ar  =  64  ? 
If  we  first  divide  by  the  co-eflicient  2  we  obtain 
x^-\-Axz=z  32. 
Then,  completing  the  square, 

a:2  -h  4a;  +  4  =  32  4-  4  =  36. 
Extracting  the  root, 

a:  -I-  2  =  ±  -v/36  =  +  6    or    -  6. 
Hence,  a?=— 2-f-6  =  +  4; 

or,  a;  =  —  2  —  6  =  ~  8. 

127.  Gire  the  general  rule  for  resolving  an  equation  of  the  8«coa^ 
degree.     What  is  the  first  step  f     What  the  second  ?     Wlmt  the  third 
Wh»t  is  the  square  root  of  the  first  member  always  ecual  to  t 

If* 


202  ELEMENTARY     ALGEBRA. 

hence,  in  this  form,  the  smaller  root  is  positive,  and  tha 
larger  negative. 

Verification, 

If  we  take  the  positive  value,  viz :  a;  =  -f  4, 

the  equation  x^  -{-  4:X  =  32 

gives  42  +  4  X  4  =  32  : 

and  if  we  take  the  negative  value  of  a:,  viz  :  x  =z  —  8, 

the  equation  x'^  -{-  4:X  =  32 

givc.s  (  _  8)2  4-  4  (-  8)  =  64  -  32  =  32  ; 

froLO.  which  we  see,   that  either  of  the  values  of  ar,  yi2 
X  =:  +4  or  x=  —  8,  will  satisfy  the  equation. 

U    What  are  the  values  of  x  in  the  equation 

3a;2  +  12a;  -  19  =  -  a;2  _  12a;  +  89  ? 
If  transposing  the  terms  we  have 

3a:2  +  a;2  +  12a;  +  12a;  =  89  +  19  : 
and  by  reducing, 

4a;2  +  24a;  =  108  ; 
and  dividing  by  the  co-efficient  of  x\ 
a;2  -f  6a;  =  27. 
Now,  by  completing  the  square, 

a;2  +  6a;  4-  9  =  36, 
extracting  the  square  root, 

a;  +  3  =  ±  ■^/^  =  -|_  6  or  ->  6  : 
hence,  a;=+6-3=+3; 

or,  ,=  _6~3  =  -9. 


XQUATIONS  OF  THE  SECOND  DEGRXX.   209 

Verification. 
If  we  take  tlie  plus  root,  the  equation 
x2  4-  6a;  =  27 
gives  (3)2-f  6(3)=27; 

and  for  the  negative  root, 

x^  +  6xz=  27 
gives         (-  9)2  -f  G  (-  9)  =  81  -  54  =  27. 
4    What  are  the  values  of  a:  in  the  equation 

a;2  _  lOx  +  15  =  ^  —  34a;  +  155. 
5 

By  clearing  of  fractions,  we  have 

5x2  _  50^  -f  75  =  a;2  -  170ar  +  775: 
by  transposing  and  reducing,  we  obtain 
4x2  4-  i20x  =  700  ; 
then,  dividing  by  the  co-efficient  of  x2,  we  have 

x2  -f-  30x  =  175  ; 
and  by  completing  the  square, 

x2  -h  30x  4-  225  =  400 ; 
and  by  extracting  the  square  root, 

a-  -f  15  =  db  -/400  =  +  20  or  -  20. 
Hence,  x  =  +  5  or  —  35. 

Verijication, 
For  the  plus  value  of  x,  the  equation 
x2  4-  30x  =  175 
|5ves  (5)2  4-  30  X  5  =  25  4-  150  =  175. 


r^         204  ELEMBNTARJ-     ALGEBRA. 

And  for  the  negative  value  of  x^  we  have 

(  -  35)2  ^  30(_  35)  -  1225  -  1050  =  175. 
5.  "What  are  the  values  of  x  in  the  equation 

6  2^4  3  ^12' 

Clearing  of  fractions,  we  have 

10a:2  _  6a;  +  9  =  96  -  8a;  -  12a;2  +  273 ; 
transposing  and  reducing, 

22a:2  4.  2a;  =  360  ; 

dividing  both  members  by  22, 

_   ,    2  360 

x^  -\ X  = . 

^22  22 

Add  I  —  j     to  both  members,  and  the  equation  becomes 

^  ^22''+W   ~   22  ^W  ' 
whence,  by  extracting  the  square  root, 

"^^22"      V   22   ^V22r 


^a 


uherefore. 


1  /360~     /  1  \2 

''~"""22"^V  22  ■^122/' 

22       V   22    ^\22/* 


^ BQUATI0N8  OP  THE  SECOND  DKOREK    205 

It  remains  to  perform  the  numerical  operations.  In  the 
Qrst  place,  -^^4-  i.-^)  must  be  reduced  to  a  single  num- 
ber, having  (22)^  for  its  denominator. 

3G0      (}Y_  360  X  22  -h  1  _  7921 
'       22  ■*■  V22/  ~  (22)2        -  (22)2  > 

extracting  the  square  root  of  7921,  we  find  it  to  be  89 ; 
therefore, 

89 


vW+(SF= 


^22- 


Consequently,  the  plus  value  of  x  is 


J_      89__88_ 
22  "^  22  ~  22  ""  '^ 


and  the  negative  value  is 

""      22"^22~       11' 

that  is,  one  of  the  two  values  of  x  which  will  satisfy  the 
proposed  equation  is  a  positive  whole  number,  and  the  other 
a  negative  fraction. 

6.  What  are  the  values  of  a;  in  the  equation 
3x^-{-2x-9  =  76. 


Ans.  ]  _. 

(  a?  =  —  5f . 


7.  What  are  the  values  of  a?  in  the  equation 
2*'  +  8i  +  7  =  ^-l-  +  197. 


^•"•iL-nA 


206  ELEMENTARY     ALGEBRA. 

8.  Whai  are  the  values  of  x  in  the  equation 

Ans,   \      ~ 

(  a;  =  —  64^. 

9.  What  are  the  values  of  x  in  the  equation 

1        4  2        '^^'^l- 

Ans,   \      ~ 

\x  =  —1\, 

10.  What  are  the  values  of  x  in  the  equation 

x^        X  _  x"^        X        \Z 

¥"^T'^y~Io"^20' 


^^^-   lL-24. 


2i. 

EXAMPLES   OP   THE   SECOND   FORM. 

1.  What  are  the  values  of  x  in  the  equation 
a;2  _  8a;  +  10  =  19. 
By  transposing, 

a;2  -  8ar  =  19  -  10  =  9, 
then  by  completing  the  square 

a:2  _  Bic  +  16  =  9  +  16  =  25, 
and  by  extracting  the  root 

a?  — 4=  ± -/25  = +  5     or     —5. 
Hence, 

a;  =  4  +  5  =  9     or     a;  =  4  —  5=—  1. 

That  is,  in  this  form,  the  larger  root  is  positive  and  thf 
lesser  negative. 


IQUATIONB  or  THE  SECOND  DEOBEX.   207 

Verification, 

If  we  take  the  positive  value  of  ar,  the  equation 

x^^&z  =  9    gives,     (9)2  -  8  X  9  =  81  -  72  =  9  ; 

and  if  we  take  the  negative  value,  the  equation 

ar2-8ar  =  9,     gives,     (  _  1)2  -  8  (  -  1)  =  1  +  8  =  9; 

from  which  we  see  that  both  values  alike  :$acxsfy  the  equi^ 
tion. 

2.  What  are  the  values  of  x  in  the  equation 


X*  X 


15  =  -  +  . 


14f. 


By  clearing  of  fractions,  we  have 

Qx"  -r  4a:  -  180  =  Sx^  +  12a;  -  177, 
and  bj  transposing  and  reducing 

3a?2  —  8ar  =  3, 
and  dividing  by  the  co-efficient  of  x\  we  oV^in 

Then,  by  completing  the  square,  we  have 

2       8^16      ,  ^  1(3      25 

^'-3"+-9-  =  ^  +  ir  =  T' 

and  by  extracting  the  square  root. 


Henco, 


.--  =  ± 


25        ,    5  5 

_=  +  _   or   --, 


"*  j^  5        ,  „  4        5  1 

'=3  +  3=  +  ^'    "    '  =  ¥-¥=-3 


208  ELEMENTARY     ALGEBRA* 

Verification, 
For  the  positive  value  of  x^  the  equation 

X^  —-  —  X  =z\ 

3 
gives  32_|.x3  =  9-8  =  l: 

and  for  the  negative  value,  the  equation 

x^ a;  =  1 

3 

/       1  \2       8  1        1.8, 

3.  What  are  the  values  of  x  in  the  equation 


^-?+7|  =  8? 


Clearing  of  fractions,  and  dividing  by  the  co-efficient  of 
«^,  we  have 

Completing  the  square,  we  have 

2       ^     j^  1        11,1       49. 
^  -3^+9=^^  +  ^=36' 

then,  by  extracting  the  square  root,  we  have 

1        ^     /49        ,7  7 

^"-3=^V36  =  +6'    ^^    -6> 
hence, 

1^79^,  17  5 

^  =  ¥+6  =  0=^^'    ^^'    ^=3-6=-0 


"\^ 


EQUATIONS     OF     THE     8  £  C  OlI^J^I^A^^      200 

Verijication. 
If  we  take  the  positive  value  of  ar,  the  equation 

..ives,  (lJ)^_|xH  =  2l-l  =  l}: 

ind  for  the  negative  value,  the  equation 

/        5  \2       2  5       25       10       45       ,, 

«'^'^^'    (-o)    -3^-0  =36  +  18=30  =  ^*- 

4.  What  are  the  values  of  x  in  the  equation 
4a2  -  2ic2  4-  2aa:  =  18ai  -  ISi^  ? 

By  transposing,  changing  the  signs,  and  dividing  by  2,  the 
equation  becomes 

x^  —  ax  =  ^a^-  9a6  +  96^  ; 
whence,  completing  the  square, 

4  4 

extracting  the  square  root, 

a 
2 

Now,  the  square  root  of   — Qab  -f  96^,    is    evidently 

3a 

--  -  36.     Therefore, 


z  =  -^  ±  v/^  -  9a6  +  962. 


a        /3a       „,  \  \x—       2a  —  36 

=  3  ^-(2--'*)'    °'       i«=-     a  +  8». 


10 


210  BLEMENTARY     ALGEBRA. 

What  will  be  the  numerical  values  of  sc^  if  we  suppose 
«  =  6  and  6  =  1  ? 

5.  What  are  the  values  of  x  in  the  equation 

1  4 

--X  —  4  —  a;2  +  2a;  —  —x'^  =  45  —  3a;2  +  4a;1 
o  5 


Sx=      7.12  )  to  within 
^''''   U=-5.73f      0.01. 

6. 

What  are  the  values  of  x  in  the  equation 

8a:2  _  14a:  +  10  =  2a;  +  34  1 

Ans, 

[x=       3. 

(a;  =  -l. 

7. 

What  are  the  values  of  x  in  the  equation 
f!  ._  30  -f-  a;  =  2a;  -  22 1 

4c 

^/15. 

U=       8. 
1  a;  =  -  4. 

8. 

What  are  the  values  of  x  in  the  equation 
a;2  -  3a;  +  y  =  9a;  +  13J  1 

Ans. 

ix=  -I, 

9.  What  are  the  values  of  x  in  the  equation 

2aa;  —  x"^  =  —2ab  —  b^'i 

2a  +  6. 


Ans.    .  . 

0. 


(  X  = 
10.  What  are  the  values  of  x  in-the  equation 

a^ -^  b^ -.  2bx -\- x^  =  '^^^1 


Ans, 


X  =    -   ^     -  (bn  4-  va^m2  -f  Z^'-m^  —  ahiA 


1QUATI0N8     OF     THE     SECOND     D3GRBB.       211 
EXAMPLES    OF    TUB    THIRD    FORM. 

1.  What  are  the  values  of  x  in  the  equation 

a;2  +  4a;  =  -  3  1 
First,  by  completing  the  square,  we  have 
a;2  +  4a;4-4  =  -3  +  4  =  l; 
and  by  extracting  the  square  root, 

ar-f  2=  ±  -/T^  +  1,    or,    -1: 
hence,    «=— 2+1  =  —  1;   or   x=  —2  —  \=  -^ 
That  is,  in  this  form  both  the  roots  are  negative. 

Verification, 

If  we  take  the  first  negative  value,  the  equation 
a;2-f-4a:— _3 
gives,  (_i)2_f_4(_i)  =  i  _4=_3. 

and  by  taking  the  second  value,  the  equation 

ar2  +  4a:  =  —  3. 
gives,  (-  3)2  -f  4(-  3)  =  9  -  12  =  -  3  : 

hence,  both  values  of  x  satisfy  the  given  equation. 

2.  What  are  the  values  of  x  in  the  equation 

_  1.  _  5ar  -  IG  =  12  +  i-a;2  +  Qx. 
Bj  transposing  and  reducing,  we  have 

then  since  the  co-eflicient  of  the  second  power  of  x  is  nega 
tive,  we  change  the  signs  of  all  the  terms,  which  gives 

x^  +  Ux=  -28, 


il2  ELEMENTARY    ALGEBRA, 

then  by  completing  the  square 

x^-\-  11a; +  30.25  =  2.25, 
hence, 

a;  4-5.5=  ±-/2.25  =  +  1.5     or     -1.5; 
consequently, 

x=  —4     or     x=  —  7, 

3.  What  are  the  values  of  x  in  the  equation 

a;2  7 

-  -—  -  2a;  -  5  =  — a;2  +  5a;  +  5. 
o  o 

Ans,    \     ~       ^ 

4.  What  are  the  values  of  x  in  the  equation 

2 
3 

-4 

6.  What  are  the  values  of  x  in  the  equation 


2a;2  +  8a;=  -2|  — —a?. 


Ans.    i 

(  X  = 


4a;*  +  -r-a;  +  3a;  =  —  14a;  —  3i  -  4a;2. 
5 


A                \   X  — 

Ans.   • 

X  =. 

-2. 

6. 

What 

are  the  values  of  x 

in  the  equation 

4a;2 
=  _+24a;  +  2. 

7. 

What 

A                {  X  = 

Ans.   • 

I  X  =z 

are  the  values  of  a;  in  the  equation 

-8. 

1 
9 

a;2  +  7a;  +  20  =  - 

|_a;2_ii^._e0. 

Ans.   ]^=- 

{  X  =z   - 

-10. 

-    8. 

EQUATIONS     OF     THE     SECOND      DEGREE.       213 

8.  What  are  the  values  of  x  in  the  equation 


•-•  1 : 


Ans.  \ 


•8 

I 

9.  What  are  the  vahies  of  x  in  the  equation 

4  11  *? 

5  4  5  '"  4 

a;=:  -  10 

10.  What  are  the  vahies  of  jt  in  the  equation 

a:  -  a:2-3  =  G^-h  1. 

\  x=-\ 

11.  What  are  the  values  of  a;  in  the  equation 

xij^\x-  90  =  -  93. 

^'"    1^=1  I. 

EXAMPLES  OF  THE  FOURTH  FORM. 

1.  What  are  the  values  of  x  in  the  equation 
ar2  -  8a:  =  -  7. 
By  completing  the  square  we  have 

a:2  -  8x  +  IG  =  -  7  -f-  16  =  9 ; 
then  by  extracting  the  square  root 

a?~4=±-/9=+3    or    — 3; 
hence, 

x=-f7    or    j;=  +  l. 

That  is.  in  this  form,  both  the  roots  are  positive. 


214  ELEMENTARY      ALGEBRA. 

Verification. 

If  we  take  the  greater  root,  the  equation 
a;2„.8a;=:-7     gives     7^  -  8  X  7  =  49  -  56  =  -  7' 
and  for  the  less,  the  equation 

a;2  -  8a:  =  -  7     gives     12_8xl=l-8=-7; 
tience,  both  of  the  roots  will  satisfy  the  equation. 

2.  What  are  the  values  of  x  in  the  equation 

40 
-  1  Ja;2  -I-  3a;  -  10  =  1  Ja:^  _  iSa;  -f-  _ . 

By  clearing  of  fractions  we  have 

-  3a;2  -I-  6a;  -  20  =  Zx^  —  ZQx  +  40  ; 

then  by  collecting  the  similar  terms 

—  6a;2  +  42.r  =  60  ; 

then  by  dividing  by  the  co-efficient  of  a:^,  and  at  the  seme 
time  changing  the  signs  of  all  the  terms,  we  have 

x^  —  lx=  —  10. 

By  completing  the  square,  we  have 

a:3  _  7a;  -f  12.25  =  2.25, 

and  by  extracting  the  square  root  of  both  members, 

a?— 3.5  =  ±  ^^.25=  +  1.5    or    -1.5; 
bence, 

ic=:  3.5  + 1.5=5,     or     a;  =  3.5  -  1.5  =  2. 


XQUATI0K8  OF  THE  SECOND  DEGREE.   215 

Verification, 

If  we  take  the  greater  root,  the  equation 
x2-7a:=-10     gives     5^  -  7  X  5  =  25 -- 35  =  -  10  ; 
aiid  if  we  take  the  lesser  root,  the  equation 
ar2  -  7x  =  -  10     gives     2^  -  7  X  2  =  4  -  14  =  -  10. 

3.   What  are  the  values  of  x  in  the  equation 
-Sx-{-2x^-rl  =  17Jx  -  2x2  _  3, 
By  transposing  and  collecting  the  terms,  we  have 

42^2  _  20Jx  =  -  4  ; 
then  dividing  by  the  co-efficient  of  z*,  we  have 
a:2  _  5ja;  _  _  1. 

By  completing  the  square,  we  obtain 

,       ,,  109  ,    .   109       144 

^^-^i^+2^=-^+2^=25-' 

and  by  extracting  the  root 

o      ^,  /144        ,    12  12 

a;2  —  21  =  ±  \  / =  H or     —    - , 

^  V   25        ^   5  5 

hence. 


12  12        1 

,  =  21  +  ^  =  5;    or,    ,  =  2|--  =  ^ 


Verification, 

If  we  take  the  greater  root,  the  equation 
ar»-5ix=-l,    gives,    5^  -  5}  X  5  =  25  -  20  =  -  1 
and  if  we  take  the  lesser  root,  the  equation 

c2-5la:=-l,givee,(--j-5}x-  =  ----=.-1 


216  ELEMENTARY      ALGEBRA. 

4.  What  are  the  values  of  x  in  the  equation 

7  ^2  7^4  4 


Ans.    \ 


5,   What  are  the  values  of  x  in  the  equation 


4^2  _       a:  4-  U  =  -  5a;2  -f  8a;? 


Ans. 


6.  What  are  the  values  of  x  in  the  equation 


ft  1  11 

_4a;2_| ^ L=_3a;2 a;_L_L? 

^20         40  20     ^40* 


7.  W  hat  are  the  values  of  x  in  the  equation 
x^-lO^\x=-ll 


(ar=10 
( ^  — 17' 


Ans.     ^    _ 

lU- 
8.  What  are  the  values  of  x  in  the  equation 

1 7x^  2a;2 

27a?  +  -r-  +  100  =  —  +  12a;  -  261 


5      '  5 

Ans. 


{  x=l. 
i  x=C). 


9.  What  are  the  values  of  x  in  the  equation 

3( 

'     \x  =  l 


8a;2  73.2 

^ 22a:  +  15=  -  -—  +  28a:  -  30? 


10.  What  are  the  values  of  x  in  the  equation 

2a;2  —  30a;  -f  3=  -  a;2  +  S^\x  -  -^? 


EQUATIONS  OF  THE  BECOND  DEGREE.   217 

Properties  of  the  Boots. 

128.  We  have  thus  far,  only  explained  the  methods  o/ 
inding  the  roots  of  an  equation  of  the  second  degree.  We 
are  now  going  to  show  some  of  the  properties  of  these  roots. 

First  form. 

129.  In  the  first  form 

x^  -h  'ipx  =  q  ; 
hence,  1st  root  x  =  —  p  -{-  -y/q  -^  p^^ 

2d  root  x-=.—  p  —  -^/q^ _p*, 

and  their  sum  =  —-  2p. 

Since,  in  this  form  q  is  supposed  positive,  the  quantity 
q-\-  p"^  under  the  radical  sign  will  be  greater  than  ^>2,  and 
hence  its  root  "will  be  greater  than  p.  Consequently,  the 
first  root,  which  is  equal  to  the  difference  between  p  and 
the  radical,  will  be  positive  and  less  than  -^q  -f  p^.  In  the 
second  root,  p  and  the  radical  have  the  same  sign ;  hence, 
the  second  root  will  be  equal  to  their  sum,  and  negative 
If  we  multiply  the  two  roots  together,  we  have 


Product  equal  to 


-p 
—  p 

-    ^/q+P^ 

+p' 

-\- p\^q -\- p"" 

—  »a 


-q- 


129i  In  the  first  form,  have  the  roots  the  same  or  contrary  signal 
Wluit  is  the  sign  of  the  first  root !  What  of  the  second  ?  Which  u 
lltp  greater  I  What  is  their  sura  equal  to?  What  is  their  prod'v" 
equal  tof 


218  ELEMENTARY      ALGEBRA. 

Hence  we  conclude, 

1st.  That  in  the  first  form^  one  of  the  roots  is  always  post 
live  and  the  other  negativt 

2d.  TJiat  Ihe  positive  root  is  numerically  less  than  the 
negative  root, 

3d.  That  ihe  sum  of  the  two  roots  is  equal  to  the  co-efficient 
of  X  in  the  second  term^  taken  with  a  contrary  sign, 

4th.  That  ihe  product  of  the  two  roots  is  equal  to  tlce 
seco7id  member^  taken  with  a  contrary  sign, 

EXAMPLES. 

1.  In  the  equation 

x^-\-x  =  20, 

we  find  the  roots  to  be  4  and  ~  5.     Their  sum  is  —  1,  and 

their  product  —  20. 

2.  In  the  equation 

x^-j-2x  =  3, 

we  find  the  roots  to  be  1  and  —  3.    Their  sum  is  equal  ta 
-  2,  and  their  product  to   —  3. 

3.  The  roots  of  the  equation 

a;2  +  ic  =  90, 

are  +  9  and  —  10.     Their  sum  is  —  1,  and  their  product 
-90. 

4.  The  roots  of  the  equation 

a;2  +  4a;=60, 

we  6  and  -r-  10.     Their  sum  is   —  4,  and  their  product  is 
^GO. 


KQUATtONB  OF  TUK  SECOND  UKORBX.   219 

Let  these  principles  be  applied  to  each  of  the  examplea 
aider  "  examples  of  tue  first  form.'* 

Second  Form, 

130.  The  second  form  is, 

^2  _  2px  =  q  ; 
and  by  resolving  the  equation  we  find 


Ist  root,  z=  -\-  p  -\-  -/y  +/* 

2d  root,  x=  -\-  p—  ^/q  +p^, 

and  their  sum  =  2p. 

In  this  form,  the  first  root  is  positive  and  the  second 
negative.     If  we  multiply  the  two  roots  together,  we  have 

(jt>+  VT+P)x  {p-  VJTy)  =  -q. 

Hence,  we  conclude, 

1st.  That  in  the  second  form^  one  of  the  roots  is  positive 
%nd  the  other  negative. 

2d.  Tliat  the  positive  root  is  numerically  greater  than  the 
negative  root. 

3d.  TTiat  the  sum  of  the  roots  is  equal  to  tlie  co-efficient  of 
^  in  the  second  term,  taken  with  a  contrary  sign. 

4th.  That  the  product  of  the  roots  is  equal  to  the  second 
memher^  taken  with  a  contrary  sign. 


130.  What  is  the  sign  of  the  first  root  in  the  second  form  ?  What  m 
»he  sign  of  the  second  ?  Whicli  is  the  greater  I  What  is  their  siuu 
M]ua1  to  f     What  is  their  prtxluct  equal  to  f 


200  ELEMENTARY     ALQ£B»A. 

EXAMPLES. 

1.  The  roots  of  the  equation 

x'-xziz  12, 
lire  +  4  and  —  3.     Their  sum  is  -f  1,  and  their  product 
-12. 

2.  The  roots  of  the  equation 

x-^  -  9j\x  =  1, 

are  +10  and  —  77:-     Their  sum  is  9j®j,  and  their  product 
is  -1. 

3.  The  roots  of  the  equation 

x^-(jx-  16, 
are   +  8  and  —  2.     Their  sum  is  +  6,  and  their  product 
is  -16. 

4.  The  roots  of  the  equation 

x^-Ux=zSO, 
are  +16  and  —  5.     Their  sum  is  +11,  and  their  product 
is  —80. 

Let  these  principles  be  applied  to  each  of  the  examples 
under  "  examples  of  the  second  form." 

Third  Form, 

131.  The  third  form  is, 

x"^  +  2px  =  —  q  'j 
and  by  resolving  the  equation,  we  find, 

1st  root,  X  =  —  p  -i-  -y/  —  ^  +  p^, 


2d  root,  X  =  —  p  —  V  —  q  +  p' 


and  their  sum  is        =  —  2p. 


XQUATION8   OK  TUB   SECOND   DEUREK.   22) 

In  this  form,  tiie  quantity  under  the  radical  being  less 
than  ^9^,  its  root  will  he  less  than  p  :  hence,  both  the  roots 
will  be  negative,  and  the  first  will  be  numerically  the  lea^^ 

If  we  multiply  the  roots  together,  we  have 

flence,  we  conclude, 

1st.   That  in  the  third  form  botli  the  roots  are  negative. 

2d.   That  the  Jirst  root  is  numerically  less  than  the  second. 

3d.  That  the  sum  of  the  two  roots  is  equal  to  the  co-efficient 
of  X  in  the  second  term^  taken  with  a  contrary  sign, 

4th.  That  the  2>roduct  of  the  roots  is  equal  to  the  second 
member,  taken  with  a  contrary  sign. 

EXAMPLES. 

1.  The  roots  of  the  equation 

a:2  -h  9a:  =  —  20, 

are   —  4  and   —  5.     Their  sum  is  —  9,  and    their  product 
-1-20. 

2.  The  roots  of  the  equation 

x^^  13a:  =  -42, 

are  —  6  and  —  7.     Their  sum  is  —  13,  and  their  produof 
-h42. 


1 3 1 1  In  the  third  form,  what  are  the  signs  of  the  roota  f  "Which  rooC 
is  the  least  f  What  ia  the  sum  of  the  roota  equal  to »  What  is  theii 
produr.t  equal  to  ? 


222  ELEMENTARY     ALGEBRA* 

3    The  roots  of  the  equation 

Q 

are  —  -—  and  —  2.     Their  sum  is  —  2f ,  and  their  product 

+  H. 

4.  The  roots  of  the  equation 

a;2  +  5^  =  —  6, 

are  —  2  and  —  3.     Their  sum  is  —  5,  and  their  product 
is  -1-6. 

Let  these  principles  be  applied  to  each  of  the  examples 
unri-er  "examples  of  the  third  form." 

Fourth  Form, 

132.  The  fourth  form  is, 

x^  —  2px  =z  —  q  'y 
and  by  resolving  the  equation  we  find, 


1st  root,  X  =  p  -{■  ^  —  q-\-  p^ 

2d  root,  x^=  p  ~  ^J  —  q-\-  p^ 

Their  sum  is  =  2p. 

In  this  form,  as  well  as  in  the  third,  the  quantity  under 
the  ladical  sign  being  less  than  p^^  its  root  will  be  less  than 
p :  hence  both  the  roots  will  be  positive,  and  the  first  will 
be  the  greater. 

If  we  multiply  the  two  roots  together,  we  have 

(;> -V  V " ;  +  v^)  X  (p -  V-^-^p')  =  -f  ^. 


EQUATIONS     OF    THE    BEC0I7D     DEOREE.       223 

Hence  we  conclude, 

1st.   Thut  in  the  fourth  form^  both  the  roots  are  positive. 

2d.   That  the  Jirst  root  is  greater  than  the  second. 

3d.  That  the  sum  of  the  roots  is  equal  to  the  co-efficient  of 
X  in  the  second  term^  taken  with  a  contrary  sign. 

4th.  Tliat  the  product  of  the  roots  is  equal  to  the  second 
member^  taken  with  a  contrary  si(jn» 

EXAMPLES. 

1.  The  roots  of  the  equation 

a;2  -  7a:  =  -  12, 

are  +  4  and  +  3.     Their  sum  is  -f-  7  and  their  product 
4-12. 

2.  The  roots  of  the  equation 

a;2  _  14a;  =  ~  24, 

are  +12  and  +  2.     Their  sum  is  -f  14  and  their  product 
I  24. 

3.  The  roots  of  the  equation 

x^  -  20a;  =  -  36, 

%re  -f-  18  and   -f-  2.     Their  sum  is   -f-  20  and  their  product 
+  36. 

4.  The  roots  of  the  equation 

x^  -  17ar  =  -  42, 
are  +  14  and   +  3.     Their  sum  is  +  17  and  their  product 
f  42. 

182«  In  the  fourth  form,  what  are  the  signs  of  the  roots  f  Which  root 
is  the  greater  t  What  is  the  sum  of  the  roots  equal  to  ?  What  Isl their 
product  equal  to  ? 


234  ELEMENTARY    ALGEBRA. 

133.  In  the  third  and  fourth  forms  the  values  of  x  some 
times  become  imaginary,  and  in  such  cases  it  is  necessary 
to  know  how  the  results  are  to  be  interpreted. 

If  we  have  q  >  js^,  thai  is,  if  the  second  member  is  greater 
than  half  the  co-efficient  of  x  squared,  it  is  plain  that  y'— ^-f-jp' 
will  be  imaginary,  since  the  quantity  under  the  radical  sign 
will  be  negative.  Under  this  supposition  the  values  of  ar, 
in  the  third  and  fourth  forms,  will  be  imaginary. 

We  will  now  show  that,  when  in  the  third  and  fourth 
forms,  we  have  q  y-  p^,  the  conditions  of  the  problem  will 
be  incompatible  with  each  other. 

134.  Before  showing  this  we  will  demonstrate  a  proposi- 
tion  on  which  the  proof  of  the  incompatibility  depends  :  viz. 

If  a  given  number  be  decomposed  into  two  parts  and  those 
parts  multiplied  together,  the  product  will  be  the  greatest  pos- 
sible when  the  parts  are  equal. 

Let  2p  be  the  number  to  be  decomposed,  and  d  the  dif 
ference  of  the  parts.     Then 

/>   -f  —  =         the  greater  part  (page  104,  Ex.  7.) 
and        p  —  —  z=         the  less  part ; 

and       p^  —  —  =  F,     their  product  (Art.  40.) 

Now,  it  is  plain  that  P  will  increase  as  d  diminishes,  and 
that  it  will  be  the  greatest  possible  when  d  z=0  :  that  is, 
p  X  P  =p^     is  the  greatest  product. 


I33i  In  "which  forma  do  the  values  of  x  become  imaginary?  When 
will  the  values  of  x  be  imaginary  ?  Why  will  the  values  of  ar  be  then 
Imagmary  ? 


KQUATI0N8  OF  THE  8UC0ND  DEGREE.   225 

Now,  since  in  the  equation 

x^  —  22)x  z=  —  q 

2p  is  the  sum  of  the  roots,  and  g  their  product,  it  folhiws 
that ./  can  never  be  greater  than  2>2.  The  conditions  of  the 
proposition,  therefore,  fix  a  limit  to  the  value  of  g,  and  if 
we  make  g  >  p^^  we  express  by  the  equation  a  condition 
which  cannot  be  fulfilled,  and  this  impossibility  is  made 
apparent  by  the  values  of  x  becoming  imaginary.  Hence, 
we  may  conclude  that, 

When  the  values  of  the  unknown  guantity  are  imdginary^ 
tlie  conditions  of  the  proposition  are  incovipatiOle  with  each 
other, 

EXAMPLES. 

1.  Find  two  numbers  whose  sum  shall  be  12  and  pro- 
duct 46. 

Let  X  and  y  be  the  numbers. 

By  the  1st  condition,     x  -f  y  =  12 ; 

and  by  the  2d,  xy  =  46. 

The  first  equation  gives 

z  =  12  -  y. 
Substituting  this  value  for  x  in  the  second,  we  have 

12y  -  y2  =  46  ; 
and  changing  the  signs  of  the  terms,  we  have 
y2-  12y  =  -46. 

!84.  What  is  the  proposition  demonstrated  in  Article  134?  Tf  the 
conditions  of  the  question  are  ''ncompatible,  how  will  the  values  of  tht 
unknown  quantity  be ! 

10* 


226  ELEMENTARY     ALGEBRA. 

Then,  b;  completing  the  square 

y^  ~  12y  +  36  =  — 46  +  36  =  -  10 


which  gives  y  —  Q  -}-  -y/  —  10, 


and  y  =  6  —  V  —  10  ; 

both  of  which  values  are  imaginary,  as  indeed  they  should 
be,  since  the  conditions  are  incomj)atible. 

2.  The  sum  of  two  numbers  is  8,  and  their  prgduct  20  • 
what  are  the  numbers  % 

Denote  the  numbers  by  x  and  y. 
By  the  first  condition, 

a?  +  y=:8; 
and  by  the  second,  xy  •=.  20. 

The  first  equation  gives 

a?  =  8  —  y. 
Substituting  this  value  of  x  in  the  second,  we  have 
8y  -  y2  =  20  ; 
changing  the  signs,  and  completing  the  square,  we  have 

2/2  -  8y  -f  16  =  -  4 ; 
and  by  extracting  the  root, 


y  =  4  4-  y--4   and    y  =  4  —  y--4. 

l^hese  values  of  y  may  be  put  under  the  forms  (Art.  106) 

y  =  4  +  2  ^/^^    and    y  =  4  —  2  -/"^^TT 

3.  What  are  the  values  of  x  in  the  equation 
a:2  +  2a;  =  -  10. 


{x--\-Z  /^l, 


BQDATI0N8  OF  THE  SECOND  DEGREE.   227 


Examples  involving  more  than  one  unknown  quantity. 

1.  Given        i  ^  4-  y  =    14  )     ^^  find  a:  and  y. 

By  transposing  y  in  the  first  equation,  we  have 
ar  =  14  —  y  ; 
and  by  squaring  both  members, 

x^=  196  — 28y  H-y2. 

Substituting  this  value  for  x^  in  the  2d  equation,  we  have 
196-28y  +  y2^y2  =  100; 
from  which  we  have 


y 


2  


14y  =  -  48  ; 


and  by  completing  the  square, 

y^  -  14y  -f  49  =  1  ; 
and  by  extracting  the  square  root, 

y-7  =  db-/T=H-l    or    -1; 
henc«,         y  =  7  +  l=8,    or    y  =  7— 1=6. 

If  we  take  the  greater  value,  we  fmd  a;  =  6  ;  and  if  w<i 
take  the  lesser,  we  find  a:  =  8. 

Verification, 
For  the  greater  value,  y  =  8,  the  equation 

X  +y   =z    14     gives       6  +    8  =  14 ; 
and  x^  -\-y^z=z  100    gives    36  -}-  64  =  100. 

For  the  value  y  =  6,  the  equation 

«   4-  y   =    14     gives       8  +    6  =    14 ; 
and  a:2  -f  y2  _  loO     gives     64  -|-  36  =  100. 

Hence,  both  sets  of  values  will  satisfy  the  given  equation. 


228  ELEMENTARY     ALGEBRA, 

2.  Given     J^~~y—       l    to  find  x  and  y, 
U2  _  y2  =  45  [  ^ 

Transposing  y  in  the  first  equation,  we  have 

a;  =  3  +  y  ; 

and  then,  squaring  both  members, 

a:2  =  9  +  6y  +  y\ 

Substituting  this  value  for  x^,  in  the  second  equation,  we 
have 

9  +  6y  +  y2-y2=:45; 

whence  we  have 

6y  =  36     and     y  =  6. 

Substi luting  this  value  of  y,  in  the  first  equation,  we  have 

ar  —  6  =  3, 

and  consequently         a:  =  3  -j-  6  =  9. 

Verification, 

X  —y  =  S     gives     9  —  6  =  3; 
iuid  a:2  —  2/2  _  45     giy^g     81  —  36  =  45. 

«•«-"    I5+S+V=40f  '°'^"^^^'^^^- 

Subtracting  the  first  equation  from  the  second,  we  have 
2y2  =  18, 
which  gives  y2  _  9^ 

and  y  =  +  3,  or  —  3. 

Substituting  the  plus  value  in  the  first  equation,  we  have 
x^-{-9x  =  22  ; 


BQUATI0N8     OF     TUB     SECOND     DEGREE.       2^0 

from  which  wo  find 

a:  = -f- 2     and     ar=  — 11. 

If  we  take  the  negative  value,  y  =  —  3,  we  have  from  the 
jfirst  equation. 

x2  -  9a;  =  22  ; 
from  which,  we  find 

a;=  +  ll     and     a;=— 2. 

Verification, 

For  CnQ  valued  y  =  +  3  and  a;  =  +  2,  the  given  equation 

x^  +  Zxy  =  22 
gives  22  +  3x2x3  =  4  +  18  =  22; 

and  for  the  second  value,  x  =  —  11,  the  same  equation 

ar2  +  3ary  =  22 

gives,     (  -  11)2  4.  3  X  -  11  X  3  =  121  -  99  =  22. 

If  now  we  take  the  second  value  of  y,  that  is,  y  =r:  —  3, 
and  the  corresponding  values  of  ar,  viz.,  a*  =  +  11,  and 
X  =  —2'j  for  a;=  +  ll,  the  given  equation 

a:2  +  3ary  =  22 
gives,        IP  +  3  X  11  X  -3  =  121  -  99  =  22  ; 
and  for  ar  =  —  2,  the  same  equation 

a;2  +  2x7/  =  22 

gives,       (  -  2)2  +  3  X  -2  X  -  3  =  4  +  18  =  22. 

arz  =  y2      (|) 

4.  Given    ^a:+y+2=7     (2)^    to  find  ar.  y,  and  «. 
a-2  +  y2  +  22  =  2l     (3) 


230  ELEMENTARY      A:^GEBRA. 

Transposing  y  in  the  second  equation,  we  have 
a;  4-  2  =  7  -  2/     (4)  ; 
then  squaring  the  members,  we  have 

«2  -j-  2a;s  +  ^2  _  49  _  i^y  _j_  yi^ 

If  now  we  substitute  for  ^xz  its  value  taken  fioin  th« 
first  equation,  we  have 

a;2  _|.  2y2  4.  ^2  _  49  _  i4y  _^  2/2  . 

and  cancelling  y^^  in  each  member,  there  results 
«2  4-  y^  +  2^  =  49  -  14y. 

But,  from  the  third  equation  we  see  that  each  member  Oi 
the  liast  equation  is  equal  to  21  :  hence 

49  -  14y  =  21, 
and  14^  =  49  -  21  =  28  ; 

hence,  y  =  —  =  2. 

Placing  this  value  for  y  in  equation  (1),  gives 

ar2  =  4; 
and  placing  it  in  equation  (4),  gives 

a;  -h  2  =  5,     and    a;  =  5  —  z. 

Substituting  this  value  of  x  in  the  previous  equation,  we 
obtain 

hz  —  2^  _  4^     or     z^  —  52?  =  —  4  ; 

and  by  completing  the  square,  we  have 

^2  _  5^  4.  6.25  =  2.5, 

and        2  — 2.5=  ± '/2y5  =  +  1.5     or     -1.5; 

hence,  «  =  2.5  -h  1.5  =  4     or     2  =  -h  2.5  -  1 .5  =  1. 


KQUAnONB      OK     THE     SECOND      DEORKS.       2151 

If  we  take  the  value 

z  =  4,     we  find     ar  =  1  : 
If  we  take  the  lesser  value 

2=1,     we  find     a:  =  4. 

3.  Given     x  -f  -yfxy -^  y   =11))^,  , 

„  o      .c.^\    to  find  T  and  y. 

and         x2  -f-       xy  +  y^  =  133  i  * 

Dividing  the  second  equation  by  the  first,  we  have 
X—  -/^  -f-    y  =    7 

but,                                  X  -f-  -/xy  +    y  =  19 

hence,  by  addition,  2ar  4-  2y  =  26 

or  a;-h    y  =  13 

and  substituting,  in  Istequa.  ^/~xy-\r  13  =  19 

or,  by  transposing  -y/^=    6 

and  by  squaring  xy  =  30. 

Equation  2d,  is  a:^  +  ary  +  y^  =  133 

and  from  the  last,  we  have  3a;y          —  108. 

Subtracting  x^  —  2xy  4-  y2=    25 

hence,  x  —  y  =  ±.      5 

but  a;  -I-  y  =          13 

hence               ar  =  9  or  4  ;  and    y  =  4  or  9. 

0.  Given  the  sum  of  two  numbers  equal  to  a,  and  the 
Fum  of  their  cubes  equal  to  c,  to  find  the  numbers. 


iX       I     5/ 
y.l    ^    y3 


•232  ELEMENTARY     ALGEBRA. 

PuUiDg       X  icz  s  +  z,     and     y  =  s  —  Zy     we  have 

a 
a  =  25,     or     s  =  ■— ; 


j  a;3  =  s3  +  Ss^z  +  Ssz'^  -f 

^  (  2/3  =  S3  -  3S22  +  3S22  _ 


hence,  b/  addition,      a:3  -f  y3  _  2^3  _|_  ^^^2  _  ^^ 
whence,  z^  =  — - —    and    z  =  do\/ 


c  — 2s3 


eT"' 


^      /c—2s^  ^  fc-2s^ 

or,  by  putting  for  s  its  value. 


Note. — What  are  the  numbers  when  a  =  5  and  c  —  35» 
What  are  the  numbers  when  a  =  9  and  c  =  243 1 

QUESTIONS. 

1.  Find  a  number  such,  that  twice  its  square,  added  to 
three  times  the  number,  shall  give  65. 

Let  X  denote  the  unknown  number.     Then  the  equation 
of  the  problem  will  be 

2x^  -\-Sx=z  65, 
whence, 

3  765   .    9  3  ^  23 

^=-4^V    2-  +  T6=-4^T' 


KQUATIONB  OF  THE  SECOND  DEOKKE 


2:i'^ 


Therefore, 

X— 4-  -r  =  5,     an<i 

4  4 


23 
4 


13 
2" 


Both  these  values  satisfy  the  proposition  i;  its  algebraic 
•wise.     For, 


jind    2 


2  X  (5)2  4-3x5  =  2x25-hl5=G5; 
13  \2 


(-f)+'-*=T-ir 


13       1G9      39       130 
2 


65. 


Remark. — If  we  wish  to  restrict  the  enunciation  to  iti 
arithmetical  sense,  we  will  first  observe,  that  when  x  is  re- 
placed by  —  X,  in  the  equation  2x^  -f  3ar  =  65,  the  sign  of 
the  second  term  Zx  only,  is  changed,  because  {—  xy  =  x^, 

3       23 

Therefore,  instead  of  obtaining  a:  =  —  —  ±  — -,    we  should 

3       23  13 

find  X  =  —  ±  — J    or  a:  =  — ,    and  x  =  —  5,  values  which 

only  differ  from  the  preceding  by  their  signs.     Hence,  we 

13 

[nay  say  that  the  first  negative  result, — ,    considered  in- 

dependently  of  its  sign,  satisfies  this  new  enunciation,  viz : 

To  find  a  number  such^  that  twice  its  square^  diminisJiBd 
hy  three  times  the  number^  shall  give  65.     In  fact,  we  have 

^       /13  V      o       13       169       39      ^^ 

Remark. — The  root  which  results  from  giving  the  plus 
sign  to  the  radical,  is,  generally,  an  answer  to  the  question 
both  in  its  arithmetical  and  algebraic  sense  ;  while  the  second 
root  i5  an  answer  to  it  !n  its  algebraic  sense  only. 
11 


23^1  ELEMENTARY     ALGEBRA. 

Thus,  in  the  example,  it  was  required  to  find  a  number 
of  which  twice  the  square  added  to  three  times  the  nutnbet 
shall  give  65.  Now,  in  the  arithmetical  sense,  added  means 
increased  ;  but  in  the  algebraic  sense  it  implies  diminution, 
when  the  quantity  added  is  negative.  In  this  sense,  the 
second  root  satisfies  the  enunciation. 

2.  A  certain  person  purchased  a  number  of  yards  of  cloth 
for  240  cents,  if  he  had  received  3  yards  less  of  the  same 
cloth  for  the  same  sum,  it  would  have  cost  him  4  cents  more 
per  yard.     How  many  yards  did  he  purchase  1 

Let     X  ■—     the  number  of  yards  purchased. 

240 
Then   will  express  the  price  per  yard. 

X 

If,  for  240  cents,  he  had  received  3  yards  less,  that  is 

X  —  3  yards,  the  price  per  yard,  under  this  hypothesis,  would 

240 
have  been  represented  by     -.     But,  by  the  enunciation, 

X  —  o 

this  last  cost  would  exceed  the  first  by  4  cents.     Tlierefore, 
we  have  the  equation 

240        240 


X 

-3         X 

-  =  t , 

} 

whence,  b; 

Y  reducing, 
3 

x'-Zxz^. 

zl80, 

and 

ty/i.4_180  = 

3  ±27 
2 

tharefore, 

X  = 

15     and 

X  —  - 

-12. 

The  value  a;  =  15  satisfies  the  enunciation  ;  for,  15  yards 

240 
for  240  cents,  gives     — — -,     or   16   cents,  for  the  price  of 
Id 

one  yard  ;  and  12  yards  for  240   cents,  gives  20  ceatsfor  the 

price  of  one  yard,  which  exceeds  16  by  4. 


EQUATIONS     or     THE     8SC0ND     DEftRSJ:.       2^5 

As  to  the  second  solution,  we  can  form  a  new  enuncie^ 
tion,  with  which  it  will  agree.  For,  going  back  to  the 
equation,  and  changing  z  into  —  a:,  we  have 

240  240       ^  240        240 


—  x  —  S       —X         '  X         a:-f3 

an  equation  which  may  be  considered  the  algebraic  transla 
tion  of  this  problem,  viz. :  A  certain  person  purchased  a  nurrv- 
her  of  yards  of  cloth  for  240  cents  :  if  he  had  jxiid  the  sains 
Bumfor  3  yards  more,  it  would  have  cost  him  4  cents  less  per 
yard.     How  many  yards  did  he  purchase? 

Ans.  X  =  12,  and  a:  =  —  15. 

3.  A  man  bought  a  hors€,  which  he  sold  for  24  dollars 
At  this  sale,  he  lost  as  much  per  cent,  upon  the  price  of  his> 
purchase  as  the  horse  cost  him.  What  did  he  pay  for  the 
horse? 

Let  X  denote  the  number  of  dollars  that  he  paid  for  the 
horse ;  then,  x  —  24  will  express  the  loss  he  sustained.    But 

X 

as  he  lost  x  per  cent,  by  the  sale,  he  must  have  lost  —r 
upon  each  dollar,  and  upon  x  dollars  he  lost  a  sum  denoted 
by    — —  ;    we  have  then  the  equation 

x^ 
— —  =  ar  —  24,     whence     rr^  —  1 00^  =  —  2400 ; 

and  ar  =  50  ±  ^^2500  -  2400  =  50  ±  10. 

Therefore,  ar  =  GO    and    x  =  40. 

Both  of  these  values  will  satisfy  the  question. 

For,  in  the  first  place,  suppose  the  man  gave  $G0  for  the 
horse  and  sold  him  for  24,  he  loses  36.  Aga'ji,  from  tho 
euuuoiation,  he   should   lose   GO  per  cent,  of  60,  that   is, 


236  EI.EMEyiARY    ALGEBRA. 

-— -  of  60,  01   — -— — ,  which  reduces  to  36  ;  therefore 

00  riatisfies  the  enunciation. 

Had  he  paid  |40,  he  would  have  lost  $16  by  the  sale ; 

40 

for,  he  should  lose  40  per  cent,  of  40,  or  40  X  -r^,    which 

reduces  to  16;  therefore,  40  verifies  the  enunciation. 

4.  A  man  being  asked  his  age,  said  the  square  root  of 
my  own  age  is  half  the  age  of  my  son,  and  the  sum  of  out 
ages  is  80  years  :  what  was  the  age  of  each  ? 

Let     X  =     the  age  of  the  father. 
y  =      that  of  the  son. 
Then  by  the  first  condition 

and  by  the  second  condition 
If  we  take  the  first  equation 

y 


and  square  both  members,  we  have 

If  we  transpose  y  in  the  second,  we  have 
x  =  80  —  y: 
from  which  wg  ftnd 

2,-  r=  -  2  rt  -/324  =  16  ; 
by  taking  the  plus  root,  which  answers  to  the  question  in 
its    arithmetical    sense       Substituting   this  value,  we   find 
e  z-  04.  J         J  Father's  age  64, 

^"'-     (Sou's  10. 


KQUATI0N8     OF    THE     SECOND     DEGREE.       2o7 

5.  Find  two  numbers,  such,  that  the  sum  of  their  pro- 
dacts  by  the  respective  numbers  a  and  b,  may  be  equal  to 
2j,  and  that  their  product  may  be  equal  to  p. 

Let  X  and  y  denote  the  required  numbers  :  we  then  have 
the  equations 

ax  +  bt/z=2Sf 
and  xy=p. 

25  —  ax 
From  the  first  v  = -. ; 

0 

whence,  by  substituting  in  the  second,  and  reducing, 
a  j2  _  2sx  z=z  —  bp. 

g  I 

Therefore,         x  =  —  db  —  y's^  —  abp, 
a         a 

and  consequently, 

5  1 


y  =  ~z^—^s^  -ahp. 

This  problem  is  susceptible  of  two  direct  solutions,  be- 
cause 8  is  evidently  >  -y/s^  —  abp ;  but  in  order  that  they 
may  be  real,  it  is  necessary  that  s2>  or  =  abp. 

Let  a  =  &  ^  1  ;  the  values  of  x  and  y  reduce  to 
a;  =  5  rfc  -^/.v^  —  p    and    y  =  s  qp  y^.s-2  —  ^;. 

"Whence  we  see,  that  the  two  values  of  x  are  equal  to 
those  of  y,  taken  in  an  inverse  order ;  which  shows,  that  if 
«  +  '>/^~-'P  represents  the  value  of  ar,  s  —  -yA^  —  p  ^ill 
represent  the  corresponding  value  of  y,  and  reciprooiill/. 

This  circumstance  is  accounted  for,  by  observing  thH»  Ui 
this  paiticilar  case,  the  equations  reduce  to 


|ar-hy=2*,  ) 
\         xy=p',   \ 


238  ELEMENTARY     ALGEBRA. 

and  then  the  question  is  reduced  to  finding  two  numbers  of 
which   their  sum  is  2s,  and  their  product  p ;    or  in  other 
words,  to  divide  a  number  2s,  into  two  such  parts,  that  their 
product  may  be  equal  to  a  given  number  p. 
Let  us  now  suppose 

2s  =  14   and  p  =  AS: 

what  will  then  be  the  values  of  x  and  y  ? 

I  a;  =  8  or  G. 
[y  =z  6  or  8. 

6.  A  grazier  bought  as  many  sheep  as  cost  him  £00,  and 
after  reserving  fifteen  out  of  the  number,  he  sold  the  re- 
mainder for  £54,  and  gained  2s.  a  head  on  those  he  sold  : 
how  many  did  he  buy  1  Ans.  75. 

7.  A  merchant  bought  cloth  for  which  he  paid  £33  155., 
which  he  sold  again  at  £2  8s.  per  piece,  and  gained  by  the 
bargain  as  much  as  one  piece  cost  him :  how  many  pieces 
did  he  buy  1  A?is.  15. 

8.  What  number  is  that,  which,  being  divided  by  the  pro- 
duct of  its  digits,  the  quotient  is  3;  and  if  18  be  added  to 
it,  the  order  of  the  digits  will  be  inverted?  Ans.  24. 

9.  To  find  a  number,  such  that  if  you  subtract  it  from  10, 
And  multiply  the  remainder  by  the  number  itself,  the  pro- 
duct shall  be  21.  Ans.  7  or  3. 

10.  Two  persons,  A  and  B,  departed  from  different  places 
at  the  same  time,  and  travelled  towards  each  other.  On 
meeting,  it  appeared  that  A  had  travelled  ]8  miles  more 
than  B  ;  and  that  A  could  have  gone  B's  journey  in  15| 
days,  but  B  would  have  been  28  days  in  performing  A's 
iburney.     How  fai  did  each  travel  1 

A  72  miles. 


••{ 


'  B  54  miles. 


EQUATIONS  OP  THE  SECOND  DEGREE.   239 

11.  There  are  two  numbers  whose  difference  is  15,  and 
half  their  product  is  equal  to  the  cube  of  the  lesser  number. 
What  are  those  numbers?  Atis.  3  and  18. 

12.  What  two  numbers  are  those  whose  sum,  multiplied 
by  the  greater,  is  equal  to  77  ;  and  who!;e  difference,  multi- 
plied by  the  lesser,  is  equal  to  121 

Ans.  4  and  7,  or  |  y^  and  y  y^. 

13.  To  divide  100  into  two  such  parts,  that  the  sum  of 
their  square  roots  may  be  14.  Ans.  04  and  36. 

14.  It  is  required  to  divide  the  number  24  into  two  such 
parts,  that  their  product  may  be  equal  to  35  times  their 
diff*erence.  Ans.  10  and  14. 

15.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their 
cubes  is  152.     What  are  the  numbers?  Ans.  3  and  5. 

IG.  Two  merchants  each  sold  the  same  kind  of  stuff*; 
the  second  sold  3  yards  more  of  it  than  the  first,  and  to- 
gether they  receive  35  dollars.  The  first  said  to  the  second, 
"  1  would  have  received  24  dollars  for  your  stuff";"  the 
other  replied,  "And  1  should  have  received  12^  dollars  for 
yours. '     How  many  yards  did  each  of  them  sell? 

1st  merchant  a;  =  15  x=i 

2d  "        y=18      ^^      y  =  S. 

17.  A  widow  possessed  13,000  dollars,  which  she  divided 
into  two  parts,  and  placed  them  at  interest,  in  such  a  man- 
ner, that  the  incomes  from  them  were  equal.  If  she  had 
put  out  the  first  portion  at  the  same  rate  as  the  second,  she 
would  have  drawn  for  this  part  300  dollars  interest ;  and  if 
she  had  placed  the  second  out  at  the  same  rate  as  the  first, 
she  would  have  drawn  for  it  490  dollars  interest.  What 
were  tlie  two  rat^s  of  interest? 

Ans.  7  and  0  per  cent. 


Ans.     < 


^140  ELEMENTARY     ALGEBRA. 


CHAPTER   VII. 

Of  Proportions  and  Progresd'.ons, 

135.  Two  quantities  of  the  same  kind  may  be  coaipaied, 
the  one  with  the  other,  in  two  ways : — 

1st.  By  considering  how  much  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference ;  and, 

2d.  By  considering  how  many  times  one  is  greater  or  less 
than  the  other,  which  is  shown  by  their  quotient. 

Thus,  in  comparing  the  numbers  3  and  12  together,  with 
respect  to  their  difference,  we  find  that  12  exceeds  3  by  9  • 
and  in  comparing  them  together  with  respect  to  their  quo- 
tient, we  find  that  12  contains  3  four  times,  or  that  12  is  4 
times  as  great  as  3. 

The  first  of  these  methods  of  comparison  is  called  Arith- 
metical  Proportion,  and  the  second.  Geometrical  Proportion. 

Hence,  Arithmetical  Proportion  considers  the  relation  of 
quantities  ivith  respect  to  their  difference,  and  Geometrical 
Proportion  the  relation  of  quantities  with  respect  to  their 
quotient. 


135i  In  how  many  ways  may  two  quantities  be  compared  the  one 
with  the  other  ?  "What  does  the  first  method  consider  ?  What  the 
second  ?  What  is  the  first  of  these  metliods  called  ?  What  is  the 
6ea>nd  called  ?     How  then  do  you  define  the  two  proportions  f 


ARITHMETICAL     P 


Of  Arithmetical  Proportion  and  Progression. 

136.  If  we  have  four  numbers,  2,  4,  8,  and  10,  of  which 
the  (liirgrence  between  the  first  and  second  is  equal  to  the 
difforence  between  the  third  and  fourth,  these  numbers  are 
said  to  be  in  arithmetical  proportion.  The  first  term  2  is 
called  an  antecedent^  and  the  second  term  4,  with  which  it  is 
compared,  a  consequcjit.  The  number  8  is  also  called  ac 
antecedent,  and  the  lunnbfr  10.  with  which  it  is  f()Mn);ircd, 
a  consequent. 

When  the  dilTerence  bLtwecn  the  fast  and  second  is  equal 
to  the  difference  between  the  third  and  fourth,  the  four  num- 
bers  are  said  to  be  in  proportion.     Thus,  the  numbers 

2,     4,     8,     10, 

are  in  arithmetical  proportion. 

137.  When  the  difierence  between  the  first  antecedent 
and  consequent  is  the  same  as  between  any  two  adjacent 
terms  of  the  proportion,  the  proportion  is  called  an  arithr- 
tnetical  progression.  Hence,  a  progression  by  differ enc3S^  or 
an  arithmetical  progression^  is  a  series  in  which  the  succes- 
sive terms  are  continually  increased  or  decreased  by  a  con 
stant  number,  which  is  called  the  common  difference  of  thf 
progression. 

Thus,  in  the  two  series 

1,     4,     7,  10,  13,  10,  19,  22,  25,  .  .  . 

60,  56,  52,  48,  44,  40,  36,  32,  28,  .  .  . 


136.  "When  are  four  numbers  in  arithmetical  proportion  I  What  if  \h%. 
first  called  f  What  is  the  second  called  f  Wliat  is  the  third  called ) 
What  ia  Uie  fourtli  called  ? 

11 


M2  ELEMENTARY     ALGEBRA 

the  fiist  is  called  an  increasing  progression^  of  which  the 
common  difference  is  8,  and  the  second  a  decreasing  pro- 
gression^ of  which  the  common  difference  is  4. 

In  general,  let  a,  h^  c^  d^  e^f,  .  .  .  designate  the  terms  of 
a  progression  by  differences ;  it  has  been  agreed  to  write 
them  thus : 

a.h.c.d.e.f.g.h.i.k... 

rhis  series  is  read,  a  is  to  J,  as  6  is  to  c,  as  c  is  to  c?,  as  d  is 
X)  e,  &;c.  This  is  a  series  of  continued  equi-differences^  in 
which  each  term  is  at  the  same  time  an  antecedent  and  a  con- 
sequent, with  the  exception  of  the  first  term,  which  is  only 
an  antecedent^  and  the  last,  which  is  only  a  consequent. 

138.  Let  d  represent  the  common  difference  of  the  pro- 
gression 

a.h.c.e.f.g.h^  &c., 

which  we  will  consider  increasing. 

From  the  definition  of  the  progression,  it  evidently  fol- 
lows that 

h  ::^a  -{-  d^     c  =  i+c?=a  +  2o?,     e  =  c  -\-  d  z=z  a-^  3c?; 

md,  in  general,  any  term  of  the  series  is  equal  to  the  first 
term  plus  as  many  times  the  common  difference  as  there  are 
preceding  terms. 

Thus,  let  /  be  any  term,  and  n  the  number  which  marks 
ihe  place  of  it :  the  expression  for  this  general  term  is 
l  =  a-\-{n  —  \)d. 


137.  What  is  an  arithmetical  progression  ?  What  is  the  number  call- 
ed hy  which  the  terms  are  increased  or  diminished  ?  What  is  an  increas- 
ing progression  ?  What  is  a  decreasing  progression  ?  Which  term  ifi 
only  an  antecedent  ?     Which  only  a  consequent  ? 


ARITHMETICAL     PROUKE&6ION.  243 

Hence,  for  finding  the  last  terui,  we  have  the  following 

RULE. 

1.  Multiply  the  common  difference  by  the  number  of  termt 
less  one. 

il  To  the  product  add  the  Jirst  term  •  the  sum  will  be  ths 
last  term, 

EXAMPLES. 

The  formula  I  =  a  -\-  {n  —  \)d  serves  to  find  any  term 
whatever,  without  our  being  obliged  to  determine  all  those 
which  precede  it. 

L  If  we  make  n  —  \^  we  have  1=  a\  that  is,  the  series 
will  have  but  one  term. 

2.  If  we  make  n  =  2,  we  have  lz=.a-{-  d\  that  is,  the 
series  will  have  two  terms,  and  the  second  term  is  equal  to 
the  first  plus  the  common  difference. 

3.  If  a  =  3  and  d  =  2,  what  is  the  3d  term  1       Ans.  7. 

4.  If  a  =  5  and  c?  =  4,  what  is  the  6th  term  1     Ans.  25. 

5.  If  a  =  7  and  d  =  5,  what  is  the  9th  term?     Ans.  47. 

6.  If  a  =  8  and  c?  =  5,  what  is  the  tenth  term  ? 

Ans.  53. 

7.  If  a  =  20  and  rf  =  4,  what  is  the  12th  term? 

Ans.  64. 

8.  If  a  =  40  and  d  —  20,   what  is  the  50th  term  ? 

Ans,  1020. 


138.  Give  tho  rule  fur  finding  the  lost  tenu  uf  a  series  when  the  pro 
greesiuu  it  iuoreaalLg. 


244  ELEMENTART      ^LGEBRA. 

9.  If  a  =  45  and  d  =  30,  what  is  the  40th  term  ? 

Atis.  1215. 

10.  If  a  =  30  and  d  =  20,  what  is  the  60th  term  1 

Ans.  1210. 

11.  If  a  =  50  aud  d=  10,  what  is  the  100th  term] 

Ans.  1040. 

12.  To  find  the  50th  term  of  the  progression 

1  .  4  .  7  .  10  .  13  .  16  .  19  .  .  ., 
we  have  Z  =  1  +  49  x  3  =  148. 

13.  To  find  the  60th  term  of  the  progression 

1  .  5  .  9  .  13  .  17  .  21  .  25  .  .  ., 
we  have  /  =  1  -f  59  X  4  =  237. 

139.  If  the  progression  were  a  decreasing  one,  we  should 
have 

I  =  a  —  {n  —  l)d. 

Hence,  to  find  the  last  term  of  a  decreasing  progression,  we 
have  the  following 

RULE. 

I.  Multiply  the  common  difference  hy  the  number  of  term* 
less  one. 

II.  Subtract  the  product  from  the  first  term  •  the  remainder 
will  be  the  last  term. 


139.  Give  the  rule  for  finding  the  last  tenu  of  a  series,  when  the  pro 
greesiuu  is  decreasing. 


ARITHMETICAL     PROGRESSION.  245 


EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  60,  the 
number  of  terms  20,  and  the  common  difference  3  :  what  it 
the  last  term  1 

l=a-(n-\)d    gives    /=:60-(20-l)3=z 60-57  =  3. 

2.  The  first  term  is  90,  the  common  difference  4,  and  the 
number  of  terms  15 :  what  is  the  last  term  1  Ans.  34. 

3.  The  first  term  is  100,  the  number  of  terms  40,  and  the 
common  difference  2 :  what  is  the  last  term  ?  A71S.  22. 

4.  The  first  terra  is  80,  the  number  of  terms  10,  and  the 
common  difference  4  :  what  is  the  last  term  1  Ans.  44. 

5.  The  first  term  is  600,  the  number  of  terms  100,  and 
the  common  difference  5  :  what  is  the  last  term  1 

Ans.  105. 

6.  The  first  term  is  800,  the  number  of  terms  200,  and 
the  common  difference  2  :  what  is  the  last  term  1 

Ans.  402. 

140.  A  progression  by  differences  being  given,  it  is  pro- 
posed to  prove  that,  the  s^im  of  any  two  terms^  taken  at  equal 
distances  from  the  two  extremes^  is  equal  to  the  sum  rf  tl^e  two 
extremes. 

That  is,  if  we  have  the  progression 

2  .  4  .  6  .  8  .  10  .  12, 

we  wish  to  prove  generally,  that 

4+10     or    6  +  8, 

ts  equaJ  to  the  sum  of  the  two  extremes  2  and  12. 


246  ELEMENTARY     ALGEBRA 

Let  a.b.c.e,f....i,k.  I  be  the  proposed 
progression,  and  n  the  number  of  terms. 

We  will  first  observe  that,  if  x  denotes  a  term  which  haa 
p  terms  before  it,  and  y  a  term  which  has  p  terms  after  it, 
we  have,  from  what  has  been  said, 

X  =z  a  -{-  p  X  d^ 
and  y  z=  I  —p  X  d] 

whence,  by  addition,    x  -^  y  =  a  -{-  I, 

which  proves  the  proposition. 

Referring  to  the  previous  example,  if  we  suppose,  in 
the  first  place,  x  to  denote  the  second  term  4,  then  y 
will  denote  the  term  10,  next  to  the  last.  If  x  denotes 
the  3d  term  6,  then  y  will  denote  8,  the  third  term  from 
the  last. 

Having  proved  the  first  part  of  the  proposition,  write  the 
terms  of  the  progression,  as  below,  and  then  again,  in  an 
inverse  order,  viz. : 

a.h.c,d.e.f,,.i.k,L 

I  ,  k  .  i c  .  b  .  a. 

Calling  S  the  sum  of  the  terms  of  the  first  progression, 
2/S  will  be  the  sum  of  the  terms  of  both  progressions,  and 
we  shall  have 

2S  =  {a  +  l)  +  {b^-k)  +  {c-^i),..-\-(i+c)  +  {k-{-b)  +  {l^d), 

Now  since  all  the  parts,  a  -{- 1^  b  -\-  k^  c  -\-  i  .  .  ,  s^yq  equal 
to  each  other,  and  their  number  equal  to  «, 

25=(a-fOXn,     or     S={^~^\xn. 


ARITHMETICAL     PICOORKS8ION.  247 

Hence,  for  finding  the  sum  of  an  arithmetical  series,  we 
hf*  >.'e  the  following 


RULE. 

I.  Add  the  two  extremes  together,  and  take  half  their  sum, 

II.  Multiply  this  half-sum  by  the  number  of  terms;    the 
product  will  be  the  sum  of  the  series. 

EXAMPLES. 

1.  The  extremes  are  2  and  16,  and  the  number  of  terms 
8  :  what  is  the  sum  of  the  series  1 


S  +  y-^j  X  «,    gives    S 


^±iix8  =  72. 


2.  The  extremes  are  3  and  27,  and  the  number  of  terms 
12  :  what  is  the  sum  of  the  series'?  Ajis.  ISO. 

3.  The  extremes  are  4  and  20,  and  the  number  of  terms 
10  :  what  is  the  sum  of  the  series  ?  Ans.  120. 

4.  The  extremes  are  100  and  200,  and  the  number  of 
terms  80  :  what  is  the  sum  of  the  series  1  Ans.  12000. 

5.  The  extremes  are  500  and  GO,  and  the  number  of  terms 
20  :  what  is  the  sum  of  the  series  ?  Ans.  5600. 

6.  The  extremes  are  800  and  1200,  and  the  number  of 
terms  50  :  what  is  the  sum  of  the  series  ?  Ans.  50000. 


140.  In  every  progreasion,  what  ia  the  sum  of  the  two  extremea 
equal  to  ?  What  ia  the  nJe  for  finding  the  aum  of  an  arithmetical 
teriea  t 


248  ELEMENTARY     ALGEBRA. 

141.  In  arithmetical  proportion  there  are  five  numbers  to 
be  considered : — 

1st.  The  first  term,  a. 

2d.    The  common  diflTerence,  d. 

3d.    The  number  of  terms,  n. 

4th.  The  last  term,  /. 

5th.  The  sura,  S. 

The  formulas 

I  =  a -\-  {n  —  \)d  and     S  =  [—^  )  X  n 

contain  five  quantities,  a,  d,  n,  I,  and  ^S^  and  consequently 
give  rise  to  the  following  general  problem,  viz  :  Any  three 
of  these  Jive  quantities  being  given,  to  determine  the  other 
two. 

We  already  know  the  value  of  >S^  in  terms  of  a,  n,  and  I. 

From  the  formula 

I  z=  a  -\-  (n  —  l)c?, 
we  find  a  =  I  —  (n  —  l)d. 

That  is :  The  first  term  of  an  increasing  arithmetical  pro- 
gression is  equal  to  the  last  term,  minus  the  product  of  the 
common  difference  by  the  number  of  terms  less  one,  . 

From  the  same  formula,  we  also  find 

I  —  a 

d= -. 

n  —  \ 

That  is :  In  any  arithmetical  progression,  the  common  differ- 
ence is  equal  to  the  last  term  minus  the  first  term  divided  by 
the  number  of  terms  less  one. 

141.  How  many  numbers  are  considered  in  arithmetical  proportion? 
What  are  they  ?  In  every  arithmetical  progression,  what  is  the  comuwa 
iliiTerenoe  equal  to  il 


ARITHMETICAL     PR00RKS8I0N.  249 

The  last  term  is  16,  the  first  term  4,  and  the  number  of 
terms  5  :  what  is  the  common  differesce  1 

The  formula  d= 


n  -  1 


givea  0  =  — - —  =3. 


2.  The  last  term  is  22,  the  llrst  term  4,  and  the  numbei 
of  terms  10  :  what  is  the  common  diflerence?  Ans.  2. 

142.  The  last  principle  affords  a  solution  to  the  following 
question : 

To  Jind  a  number  m  of  arithmetical  means  between  two 
given  numbers  a  aiid  b. 

To  resolve  this  question,  it  is  first  necessary  to  find  the 
common  difference.  Now,  we  may  regard  a  as  the  first 
term  of  an  arithmetical  progression,  b  as  the  last  term,  and 
the  required  means  as  intermediate  terms.  The  number  of 
terms  of  this  progression  will  be  expressed  by  7?i  -f  2. 

Now,  by  substituting  in  the  above  formula,  b  for  /,  and 
m  -f  2  for  n,  it  becomes 

b  —  a      b  —  a  ^ 

-m'+2-  l~m-f  1' 

that  is  :  TVie  common  difference  of  the  required  progression  it 
obtained  bg  dividing  the  difference  between  the  given  mimben 
a  and  b,  bg  the  required  number  of  means  plus  one. 


142.  How  do  you  find  any  number  of  arithmetical  meana  between 
two  given  iiumbers  t 

11* 


250  ELEMENTARY     AI.  GEBRA. 

Having  oLtained  the  common  difTerence,  form  the  second 
term  of  the  progression,  or  the  first  arithmetical  mean,  by 
adding  d  to  the  first  term  a.  The  second  mean  is  obtained 
by  augmenting  the  first  mean  by  c?,  &c. 

1.  Find  three  arithmetical  means  between  the  extremes 
2  and  18. 

b  —  a 


The  formula  d  ■■ 


m+  1 


gives  a  =  — - — =4; 


hence,  the  progression  is 

2  .  6  .  10  .  14  .  18. 

2.  rind  twelve  arithmetical  means  between  12  and  77 
b  —  a 


The  formula  d  =^ 


m-f-  1 

^     77-12      . 
gives  d  =  — — — =5. 

Hence,  the  progression  is 

12  .  17  .  22  .  27  ....  77. 

143.  Remark. — If  the  same  number  of  arithmetical 
means  are  inserted  between  all  the  terms,  taken  tv/o  and 
two,  these  terms,  and  the  arithmetical  means  united,  will 
form  but  one  and  the  same  progression. 

For,  let  a  .  b  .  c  .  e  .  f  ,  .  .  be  the  proposed  pro- 
gression, and  m  the  number  of  means  to  be  inserted  be- 
tween  a  and  b,  b  and  c,  c  and  e  ,  ,  .  .    6zni, 


ARITHMETICAL     PROGRESSION.  251 

From  what  has  just  been  said,  the  common  difference  ol 
each  partial  progression  will  be  expressed  by 

b  —  a        c  —  b        e  —  c 

m-f  1'      m  -h  V      m+"T  *  *  * 

expressions  which  are  equal  to  each  other,  since  a,  b,  e  .  ,  , 
are  in  progression  :  therefore,  the  common  difference  is  lh« 
same  in  each  of  the  partial  progressions;  and  since  the 
la^t  term  of  the  first,  forms  tYiQ^rsi  term  of  the  second,  (fcc, 
we  may  conclude  that  all  of  these  partial  progressions  form 
a  single  progression. 


EXAMPLES. 

1.  Find  the  sum  of  the  first  fifty  terms  of  the  progres- 
slon   2.9.   IG  .  23  .  .  .* 

For  the  50th  term  we  have 

/=2-f-49  X  7  =  345. 


50 
Hence,    ^  =  (2  -f  345)  X  17  =  347  X  25  =  8675. 


2.  Find  the  100th  term  of  the  series  2  .  9  .  16  .  23  .  .  . 

Ans.  G95. 

3.  Find  the  sum  of  100  terms  of  the  series  1.3.5. 
r  .  9  .  .  .  Ans.  10000. 

4.  The  greatest  term  is  70,  the  common  difference  3,  and 
the  number  of  terms  21  :  what  is  the  least  term  and  the 
6um  of  the  seues? 

Ans.  Least  term  10 ;  sum  of  series  840. 


252  ELEMENTARY     ALGEBEA. 

5.  The  first  term  is  4,  the  common  difference  8,  and  the 
number  of  terms  8  :  what  is  the  last  term,  and  the  sum  of 
the  series  1 

Last  term  60. 

Sum     =  256. 


A71S.    \ 


6.  The  first  term  is  2,  the  last  term  20,  and  the  number 
of  terms  10  :  what  is  the  common  difference  1 

Ans.  2. 

7.  Insert  four  means  between  the  two  numbers  4  and  19 : 
what  is  the  series  ? 

A71S.  4  .  7  .  10  :   13  .   16  .  19. 

8.  The  first  term  of  a  decreasing  aritlimetical  progression 
is  10,  the  common  difference  one-third,  and  the  number  of 
terms  21  :  required  the  sum  of^he  series. 

Ans.  140. 

9.  In  a  progression  by  differences,  having  given  the  com- 
mon difference  6,  the  last  term  185,  and  the  sum  of  the 
terms  2945  :  find  the  first  term,  and  the  number  of  terms. 

Ans.  First  term  =  5  ;  number  of  terras  31 

10.  Find  nine  arithmetical  means  between  each  antece- 
dent and  consequent  of  the  progression  2.5.8.11.  14... 

Ans.  Common  dif ,  or  d  =  0.3, 

11.  Fhid  the  number  of  men  contained  in  a  triangular 
battalion,  the  first  rank  containing  one  man,  the^  second  2, 
the  third  3,  and  so  on  to  the  /i'*,  which  contains  n.  In  other 
words,  find  the  expression  for  the  sum  of  the  natural  num 
bers  1,  2,  3  .  .  .,  from  I  to  n  inclusively. 


QKOMETRICAL     PROPORTION.  253 

12.  Find  the  sum  of  the  n  first  terms  of  the  progression 
of  uneven  numbers  1,  3,  5,  7,  9  .  .  .  Ans.  S  =  li^. 

13.  One  hundred  stones  being  placed  on  the  ground  in  a 
straight  line,  at  the  distance  of  2  yards  apart,  how  far  will  a 
person  travel  who  shall  bring  them  one  by  one  to  a  basket, 
placed  at  a  distance  of  2  yards  from  the  first  stone  ? 

Ajis.  11  miles,  840  yards. 

Oeometrical  Proportion  and  Progression. 

144.  liaiio  is  the  quotient  arising  from  dividing  one 
quantity  by  another  quantity  of  the  same  kind.  Thus,  if 
the  numbers  3  and  6  have  the  same  unit,  the  ratio  of  3  to  6 
will  be  expressed  by 

And  in  general,  if  A  and  B  represent  quantities  of  the  same 
kind,  the  ratio  of  ^  to  -S  will  be  expressed  by 

B 

A' 

146.  If  there  be  four  numbers 

2,     4,     8,     16, 

having  such  values  that  the  second  divided  by  the  first  is 
iqual  to  the  fourth  divided  by  the  third,  the  numbers  are 

144.  What  is  ratio  ?     What  k  tJic  ratio  of  3  to  6  ?     Of  4  to  12  t 


254  ELEMENTARY    ALGEBRA. 

said  to  be  in  proportion.  And  in  general,  if  there  be  f(>u! 
quantities,  A,  B,  (7,  and  D,  having  such  values  that 

A-  C 

then  A  is  said  to  have  the  same  ratio  to  B  that  C  has  to  />, 
or,  the  ratio  of  A  to  B  is  equal  to  the  ratio  of  C  to  D. 
When  four  quantities  have  this  relation  to  each  other,  com- 
pared together  two  and  two,  they  are  said  to  be  in  geomet- 
rical proportion. 

To  express  that  the  ratio  of  ^  to  -5  is  equal  to  the  ratio 
of  C  to  2>,  we  write  the  quantities  thus  : 

A  :  B  ::   C  :  B  ; 

and  read,  ^  is  to  ^  as  C  to  J), 

The  quantities  which  are  compared,  the  one  with  the 
other,  are  called  terms  of  the  proportion.  The  first  and  last 
terms  are  called  the  two  extremes,  and  the  second  and  third 
terms,  the  two  means.  Thus,  A  and  J)  are  the  extremes, 
and  B  and  C  the  means. 

146.  Of  four  proportional  quantities,  the  first  and  third 
are  called  the  antecedents,  and  the  second  and  fourth  the 
consequents  ;  and  the  last  is  said  to  be  a  fourth  proportional 
to  the  other  three,  taken  in  order.  Thus,  in  the  last  pro- 
portion  A  and  0  are  the  antecedents,  and  B  and  D  the  con- 
sequents. 

145.  "What  is  proportion  ?  How  do  you  express  that  four  numbers 
are  in  proportion  ?  What  are  the  numbers  called  ?  What  are  the  first 
and  fourtli  terms  called  ?     What  the  second  and  third  ? 

146.  In  four  proportional  quantities,  what  are  the  nrsf  and  third  called  I 
What  the  eecuud  and  fourth  ? 


OKOMKTRICAL     PROPORTION.  255 

147.  Three  quantities  are  in  proportion  when  the  first  has 
the  same  ratio  to  the  second  that  the  second  has  to  the 
third ;  and  then  the  middle  term  is  said  to  be  a  mean  pro- 
portional between  the  other  two.     For  example, 

3  :  6  :  :  6  :  12; 

and  6  is  a  mean  proportional  between  3  and  12. 

148.  Qiiantitie^s  are  said  to  be  in  proportion  by  inversion^ 
or  inversely  J  when  the  consequents  are  made  the  antecedents 
and  the  antecedents  the  consequents. 

Thus,  if  we  have  the  proportion 

3  :  6  :  :  8  :  IG, 
the  inverse  proportion  would  bo 

6  :  3  :  :  IG  :  8. 

149.  Quantities  are  said  to  be  in  proportion  by  alterna- 
tion^ or  alternately,  when  antecedent  is  compared  with  ant& 
cedent  and  consequent  with  consequent. 

Thus,  if  we  have  the  proportion 

3  :  6  :  :  8  :  IG, 

the  alternate  proportion  would  be 

3  :  8  :  :  6  :  IG. 


147t  When  are  three  quantities  proportional!  What  ia  the  mlddlii 
oce  called  ? 

148.  When  are  quantities  said  to  be  in  proportion  by  inversion,  or  i» 
r^rscly  ? 

140.  When  are  quautitici  in  proportion  bj  alternation  f 


256  ELEMENTARY    ALaEBRA. 

150.  Quantities  are  said  to  be  in  proportion  by  composi- 
tion, when  the  sum  of  the  antecedent  and  consequent  is 
compared  either  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

2  :  4  :  :  8  :  16, 
the  proportion  by  composition  would  be 

2  +  4  :  4  :  :  8  +  16  :   16; 
that  is,  6:4::  24  :  16. 

151.  Quantities  are  said  to  be  in  proportion  by  division, 
when  the  difference  of  the  antecedent  and  consequent  is 
compared  either  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

3  :  9  :  :  12  :  36, 

the  proportion  by  division  will  be 

9-3  :  9  :  :  36-12  :  36; 
fctat  is,  6  :  9  :  :  24  :  36. 

152.  Equi-multiples  of  two  or  more  quantities  are  the 
products  which  arise  from  multiplying  the  quantities  by  the 
same  number. 

Thus,  if  we  have  any  two  numbers,  as  6  and  5,  and  mul- 
tiply them  both  by  any  number,  as  9,  the  equi-multiples 
will  be  54  and  45  ;  for 

6  X  9  =  54,     and     5x9  =  45. 


IfiO    When  are  quantities  in  proportion  by  composition  ? 
15 1«  When  are  quantities  in  proportion  by  division  ? 
15Si  What  are  equi-multiplea  (»f  two  or  more  quantities  ? 


GEOMETRICAL     PROPORTION.  257 

m  X  ^1  and  m  x  B  aro  equi-multiples  of  A  aad  B,  the 
>mmon  multiplier  being  m. 


^ftalu 


153.  Two  quantities  ^  and  B,  which,  may  change  their 
lues,  are  reciprocally  or  inversely  proportional,  wlien  one  it 
portional  to  unity   divided  by   the  other,  and  then  their 
product  remains  constant. 

§We  express  this  reciprocal  or  inverse  relation  thus : 
■ 


^oo^ 


which  A  is  said  to  be  inversely  proportional  to  B, 

164.  If  we  have  the  proportion 

A  :  B  ::   C  :  D, 


B       D 

we  have,  -—  =  — ,    (Art.  145); 

And  by  clearmg  the  equation  of  fractions,  we  have 

BC=AD. 

That  is,  Of  four  proportional  quantities,  the  product  of  thi 
two  extremes  is  equal  to  the  product  of  the  two  means. 

This  general  principle  is  apparent  in  the  proportion  b<v 
tween  the  numbers 

2  :  10  :  ;  12  :  60, 
which  gives         2  X  GO  =  10  X  12  =  120. 


163t  Wlieii  are  two  quantities  said  to  be  reciprccally  proportional  ? 
1D4.  If  four  quantities  are  proportioual,  what  is  the  product  of  tL« 
iwu  lueoiui  equal  to  ? 

12 


258  SLBMENTART     ALGEBRA. 

155.  If  four  quantities,  A,  B^  ^j-^)  are  so  related  to  each 
other,  that 

AxI>  =  Bx  C, 

we  shall  also  have  —  =  — - ; 

-4        G 

and  hence,  A  \  B  '.  '.   0  \  D, 

That  is  :  If  the  product  of  two  quantities  is  equal  to  the  pro^ 
duct  of  two  other  quantities^  two  of  them  may  be  made  the  ex^ 
trcmes^  and  the  other  two  the  means  of  a  proportion. 
Thus,  if  we  have 

2  X  6  =-  4  X  4, 
we  also  have 

2  :  4  :  :  4  :  8. 

156.  If  we  have  three  proportional  quantities 

A  :  B  ::  B  :   C, 

B        C 
we  have,  'J  ~  B' 

hence,  B^  -  AC, 

That  is :  ^  three  quantities  are  proportional^  the  square  of 
the  middle  term  is  equal  to  the  product  of  the  two  extremes. 
Thus,  if  we  have  the  proportion 

3  :  6  :  :  6  :  12, 

we  shall  also  have 

6  X  6  =  62  =  3  X  12  =  30. 

155i  If  the  product  of  two  quantities  is  equal  to  the  product  of  tn^o 
other  qiiantities,  may  the  four  be  placed  in  a  proportion  ?     Hew  ? 

156.  If  three  quantities  are  proportional,  what  is  the  product  of  ibfii 
extrcraea  eqxial  tu  t 


OEOMBTRICAL     PROPORTION.  250 

157.  If*  we  have 

D         7) 

A  :  £  :  :   C  :  D^  and  consequently   -—  =  -^, 

Q 

multiply  both  members  of  the  last  equation  by   -5,    and 

we  then  obtain, 

C  _D 
A-B' 

Bnd  hence,  A  :   C  :  :  B  :  D. 

That  is :  If  four  quantities  are  proportional^  they  will  be  in 
proportion  by  alternation. 

Let  us  take,  as  an  example, 

10  :  15  :  :  20  :  30. 
We  shall  have,  by  alternating  the  terms, 

10  :  20  :  :  15  :  30. 

158.  If  we  have 

A  :  B  ::   C  :  I)    and     A      B  :  :  U  :  F, 

we  shall  also  have 

B       D        ^    B       r 
A  =  C    ^''^    A^-e'' 

D       F 
hence,  -;:^  =  —   and    C  \  D  w  E  \  F, 

That  is:    If  there  are  tico  sets  of  proportions  having  an 


157t  If  four  qoantities  are  proportional,  will  they  be  in  proportioo  bj 
alteniatioQ  f 


200  KLEMENTART     ALGEBRA. 

antecedent  and  consequent  in  the  one  equal  to  an  antecedenX 
and  consequent  of  the  other^  the  remaining  terms  will  he  pro- 
poriional. 

If  we  have  the  two  proportions 

2  :  6  :  :  8  :  24    and     2  :  6  :  :  10  :  30, 
we  shall  also  have 

8  :  24  :  :  10  :  30. 

159.  If  we  have 

»         7) 

A  :  B  :  :   C  :  J),    and  consequently.    —  =  —, 

^         G 

we  have,  by  dividing  1  by  each  member  of  the  equation 

A        C 

^  =  —  ,  and  consequently    £  :  A  :  :  D  :   C, 

^hat  is:  Four  proportional  quantities  will  be  in  proportion^ 
when  taken  inversely. 

To  give  an  example  in  numbers,  take  the  proportion 
7  :  14  :  :  8  :  16 ; 
then,  the  inverse  proportion  will  be 

14  :  7  :  :   10  :  8, 
in  which  the  ratio  is  one-half. 

160.  The  proportion 

A  :  B  ::   O  :  D     gives     AxD  =  Bx  C, 


158  If  you  have  t\*o  sets  of  proportions  having  an  antecedeut  and 
oonsequent  in  each,  equal ;  wliat  will  follow  ? 

ISDi  \\  four  quantities -are  in  proportion,  will  they  be  in  proportiDO 
Aen  taken  inversfcly  ? 


OKOMKTRIOAL  PROPORTION.        201 

To  each  member  of  the  last  equation  add  £  X  D,     We 
fihall  then  have 

(A-\-  B)  xD={C-[-D)x  B] 
and  by  separating  the  factors,  we  obtain 

A-\-B  :  B  :  :   C-\-D  :  D. 

^  If,  instead  of  adding,  we  subtract  B  x  D  from  both  mem- 
hei3,  we  have 

{A-B)  xD={C-I))  xB; 
which  gives 

A-B  :  B  :  :   C-D  :  D. 

That  is:  If/our  quantities  are  projiorlional^  they  will  he  in 
projmrtion  by  comj)osition  or  division. 

Thus,  if  we  have  the  proportion 

\)  :  27   :   :   IG  :  48, 
we  shall  have,  by  compositi(jn, 

9  +  27  :  27  :  :  lG  +  48  :  48; 
that  is,  36  :  27  :  :  G4  :  48, 

b  which  the  ratio  is  three-fourths. 

The  same  proportion  gives  us,  by  division, 
27-9  :  27  :  :  48-  10  :  48; 
that  is,  18  :  27  :  :  32  :  48, 

in  which  the  ratio  is  one  and  one-half. 


ICO.  If  four  quantities  are  in  proportion,  will  they  bo  in  proportion 
bv  composition  t  Will  they  be  iu  proportion  by  division  I  What  is  the 
diflhrence  between  composition  and  division ! 


262  ELEMENTARY     ALGEBRA. 

161.  If  we  have 

A~  C 

and  multiply  the  numerator  and  denominator  of  the  first 
member  by  any  number  w,  we  obtain 

— -  —  — -  and   mA  :  mB  :   :    C  \  D. 

mA       C 

That  is :  Equal  multiples  of  two  quantities  have  the  same 
ratio  as  the  quantities  themselves. 

For  example,  if  we  have  the  proportion 

5  :  10  :  :   12  :  24, 

and  multiply  the  first  antecedent  and  consequent  by  6,  we 

have 

30  :  60  :  :  12  :  24, 

in  which  the  ratio  is  still  2. 

162.  The  proportions 

A  '.  B  '.  '.   C  '.  D   2indi   A  :  B  '.  :  E  :  F, 
give         Ax  I)  =  B  X  C  and   Ax  E=B  x  E; 
adding  and  subtracting  these  equations,  we  obtain 
A{I)±F)=B{C±E),  or  A  :  B  :  :   C±E  :  B^F. 

That  is:  If  C  and  D,  the  antecedent  and  consequent^  be  aug- 
mented or  diminished  by  quantities  E  and  F,  which  have  the 
same  ratio  as  C  to  D,  the  resulting  Quantities  will  also  have 
the  same  ratio. 

161.  Have  equal  niultiples  of  two  quantities  the  same  ratio  aa  the 
quantities  ? 

162.  Suppose  the  antecedent  and  consequent  be  augmented  ordimm 
!Bb*'d  by  quantities  having  the  same  ratio  ? 


EOMETRICAL     PROPORTION. 


263 


I 


t  us  take,  as  an  example,  the  proportion 
9  :  18  :  ;  20  :  40, 
in  which  the  ratio  is  2. 

If  we  augment  the  antecedent  and   consequent   by  the 
umbers  15  and  30,  which  have  the  same  ratio,  we  shall 
have 

9+  15  :   18  4-30  ::  20  :  40; 
that  is,  24  :  48  :  :  20  :  40, 

in  which  the  ratio  is  still  2. 

If  we  aiminish  the  second  antecedent  and  consequent  by 
these  numbers  respectively,  we  have 

9  :   18  ::  20-15  :  40-30; 

tliat  is,  9  :   18  :  :  5  :  10, 

in  which  the  ratio  is  still  2. 


163.  If  we  have  several  proportions 

A  :  B  :  :   C  :  D,    which  gives   A  X  D=  B  x  C, 
A  :  B  :  :  E  '.  F;       "        "        A  x  F  =z  B  x  JS, 
A  :  B  :  :   G:  JI,        "        "        A  x  H  =  B  x  O, 
A:c.,  &c., 
we  shall  have,  by  addition, 

A(,D-irF-\-H)=zBi,C  +  E+0)\ 

and  by  separating  the  factors, 

A  '.  B  :  :   C ^  E ■\-  O  :  D+  F+  H, 

That  is  :  In  anij  number  of  proportions  having  the  samt 
iilio^  any  antecedent  will  be  to  its  consequent^  as  Vi^  snm  of 
iJie  antecedents  to  tJte  sum  of  the  consequents. 


864  ELEMENTARY     ALGEBRA. 

Let  us  take,  for  example, 

2  :  4  :  :  6  :   12     and     1  :  2  :  :  3  :  6,     &c 
Then  2  :  4  :  :  6  +  3  :  12 -f  0  ; 

that  is,  2  :  4  :  :  9  :  18, 

in  which  the  ratio  is  still  2. 

164.  If  we  have  four  proportional  quantities 

/?         T) 

A  :  B  :  :    C  :  Df    we  have     —  =  — - ; 

and  raising  both  members  to  any  power  whose  exponent  ia 
n,  or  extracting  any  root  whose  index  is  n,  we  have, 

^"       />" 

—  =z  — ,     and  consequently 

A"*  :  B"  :  :    (7"  :  i>». 
That  is  :  If  four  qna7itities  are  proportional^  their  like  powers 
or  roots  will  be  proportional. 
If  we  have,  for  example, 

2    :  4    :  :  3  . :  6, 
we  shall  have  2^  :  4^  :  :  3^  :  6^  ; 

that  is,  4    :  16  :  :  9    :  36, 

in  which  the  terms  are  proportional,  the  ratio  being  4. 

165.  Let  there  be  two  sets  of  proportions, 

n  T\ 

A  :  B  :  :   C  :  D      which  gives     —=  —^ 

A         C 

F       II 

103.  In  any  number  of  proportions  havinjj  the  same  ratio,  bow  wilJ 
any  one  antecedent  be  to  its  consequent  % 

164i  In  four  proportional  quantities,  how  are  like  powers  or  rotg? 


OKOMKTRICAI.     PROGRESSION.  265 

Mulliply  them  together,  member  by  member,  we  have 
:^  =  ^.     >^h.ch  gives     A£  :  BF  ::   CG  :  DH, 

That  is:  In  two  sets  of  proportional  quantities^  the producUt 
of  the  corresponding  terms  are  proper tiojial. 

Thus,  if  we  hnve  the  two  proportions 
8  :   10  :  :   10  :     20 
and  3  :     4  :  :     6  :       8, 

we  shall  have  24  :  04  :  :  «0  :  100. 

Oeometrical  Progresssion. 

166.  We  have  thus  far  only  considered  the  case  in  which 
the  ratio  of  the  first  term  to  the  second  is  the  same  as  that 
of  the  third  to  the  fourth. 

If  we  have  the  farther  condition,  that  the  ratio  of  the 
second  term  to  the  third  shall  also  be  the  same  as  that  of 
the  first  to  the  second,  or  of  the  third  to  the  fourth,  we  shall 
have  a  series  of  numbers,  each  one  of  which,  divided  by  the 
preceding  one,  will  give  the  same  ratio.  Hence,  if  any 
term  be  multiplied  by  this  quotient,  the  product  will  be  the 
succeeding  term.  A  series  of  numbers  so  formed  is  called 
B  geometrical  progression.     Hence, 

A  Geometrical  Progression^  or  progression  hy  quotients^  is  9 
sei'ies  of  terms,  each  of  which  is  equal  to  the  preceding  term 


1C5.  In  two  8ct8  of  proportiouA,  how  are  the  producta  of  the  correi 
PolkUum  terms  t 


26(5  ELEMENTARY     ALGEBRA. 

multiplied  by  a  constant  number^  which  number  ia  called  the 
ratio  of  the  progression.     Thus, 

1  :  3  :  9  :  27  :  81   :  243,  &c., 

is  a  geometrical  progression,  in  which  the  ratio  is  3.  It  U 
w^ritten  by  merely  placing  two  dots  between  the  terms. 

AJso,  64  :  32  :   16  :  8  :  4  :  2  :   1 

.  is  a  geometrical  progression,  in  which  the  ratio  is  one-half. 

In  the  first  progression  each  term  is  contained  three  times 
in  the  one  that  follows,  and  hence  the  ratio  is  3.  In  the 
second,  each  term  is  contained  one-half  times  in  the  one 
which  follows,  and  hence  the  ratio  is  one-half. 

The  first  is  called  an  increasing  progression,  and  the 
second  a  decreasing  progression. 

Let  a,  6,  c,  c?,  e,  /,  .  .  .  be  numbers  in  a  progression  by 
quotients  ;  they  are  written  thus  : 

a  -.  h  '.  c  '.  d  '.  e  \  f  '.  g  ,  ,  , 

and  it  is  enunciated  in  the  same  manner  as  a  progression  by 
differences.  It  is  necessary,  however,  to  make  the  distinc- 
tion, that  one  is  a  series  formed  by  equal  differences,  and 
the  other  a  series  formed  by  equal  quotients  or  ratios.  It 
should  be  remarked  that  each  term  is  at  the  same  time  an 
antecedent  and  a  consequent,  except  the  first,  which  is  only 
an  antecedent,  and  the  last,  which  is  only  a  consequent. 


166 1  What  is  a  geometrical  progression  ?"  "What  is  the  ratio  of  th« 
progression  ?  If  any  term  of  a  progression  be  multiplied  by  the  ratio 
what  will  the  product  be  ?  If  any  term  be  divided  by  the  ratio,  what 
will  the  quotient  be  ?  How  is  a  progression  by  quotients  written  \ 
Which  of  the  terms  is  only  an  antecedent  ?  Which  only  a  consequent  I 
How  may  each  of  the  others  be  considered  ? 


*  OBOMElKlCAL     PROORE88ION.  267 

167.  Let  q  denote  the  ratio  of  the  progression 

a  '.  b  :  c  '.  d  .  .  .  '^ 

f  being  >  1  when  the  progression  is  increasing^  and  9  <  1 
when  it  is  decreasing.     Then,  since 

h  c  d  e  . 

T  =  *'    T  =  ^'    ~  =  ^'     ■rf=?>  *''=•• 

we  have 

b  —  aq^     e  =  bg  z=  aq^^     d  =r.  cq  =  aq^      e  =  dq  =  aq*j 
/=  eq  =  aq^  .  .  . ; 

that  is,  the  second  term  is  equal  to  aq,  the  third  to  aq"^^  the 
fourth  to  aq^j  the  fifth  to  a^*,  &c. ;  and  in  general,  the  7ith 
term,  that  is,  one  which  has  n  —  \  terms  before  it,  is  ex- 
pressed by   aq*~^. 

Let  /  be  this  term ;  we  then  have  the  formula 

by  means  of  which  we  can  obtain  any  term  without  being 
obliged  to  find  ail  the  terms  which  precede  it.  Hence,  to 
ftud  the  last  term  of  a  progression,  we  have  the  following 

RULE. 

L  liaise  the  ratio  to  a  power  whose  exponent  is  one  less  than 
the  number  of  terms, 

II.  MuUiplij  the  power  thus  found  by  the  first  term :  tli4 
product  will  be  the  required  term. 

167.  By  what  letter  do  we  Jenote  the  ratio  of  a  progressior  f  In  an 
•jicreasiiig  progression  is  q  greater  or  less  than  If  In  a  decreasing  pro- 
fession is  q  greater  or  less  than  1  ?  If  a  is  the  first  term  aiul  q  the 
ratio,  what  is  tl)e  seconJ  ♦em  equal  to?  What  the  third t  What  the 
tuurth !  What  is  the  Ias  «erm  equal  to  ?  Give  the  rule  for  finding  the 
last  Urm. 


268  ELEMENTARY      ALGEBRA.  * 

EXAMPLES. 

1.  Find  the  5th  term  of  the  progression 

2  :  4  :  8  :  16  .  . 
in  which  the  first  term  is  2  and  the  common  ratio  2 
5th  term  =  2  x  2*  =  2  X  16  =  32     Ans 

2.  Find  the  8th  term  of  the  progression 

2  :  6  :  18  :  54  .  .  . 

8th  term  =  2  X  3^  =  2  X  2187  =  4374     Ans. 

3.  Find  the  6th  term  of  the  progression 

2  :  8  :  32  :  128  .  .  . 
6th  term  =  2  x  4^  =  2  X  1024  =  2048     Ans. 

4.  Find  the  7th  term  of  the  progression 

3  :  9  :  27  :  81  .  .  . 

7th  term  =  3  X  3^  =  3  x  729  =  2187     Ans 

5.  Find  the  6th  term  of  the  progression 

4  :   12  :  36  :  108  .  .  . 
6th  term  =  4  X  3^  =z  4  X  243  =  972     Ans. 

6.  A  person  agreed  to  pay  his  servant  1  cent  for  the  first 
day,  two  for  the  second,  and  four  for  the  third,  doubling 
every  day  for  ten  days:  how  much  did  be  receive  on  the 
tenth  day  1  Ans.  15.12. 


•         GEOMETRICAL  PROORKBhION.        2»»9 

7.  What  is  the  8th  term  of  the  progression 

ft  :  36  :   144  ;  576  .  .  . 
8th  term  =  9  x  4''  =  9  x  16384  =  147456    Anr 

8.  Find  the  12th  term  of  the  progression 

64  :  16  :  4  :  1  :  —  .  .  . 
4 


12th 


term  =  64  (-j     =  —  =  -  =  ---     Ans, 


168.  We  will  now  proceed  to  determine  the  sum  of  n 
t<^rms  of  a  progression 

a  :  b  :  e  :  d  :  e  :  f  :  .  .  ,  :  i  :  k  :  I; 

I  denoting  the  nth  term. 

We  have  the  equations  (Art.  167), 

b  =  aq,     c=:.hq,     d=:cq^     e=^dq^  .  .  .  k-=iiq^     lz=:kq, 

and  by  adding  them  all  together,  member  to  member,  we 
deduce 

8wn  0/  1st  memherB,  Sum  of  2d  memhen. 

h-{-c-\-d-\-e-\-  .  .  .   -\-k-\-l—{a  +  b-\-c-\-d-\-  .  .  .  -\-i-^k)q' 

in  which  we  see  that  the  first  member  contains  all  the  term» 
but  a,  and  the  polynomial  within  the  parenthesis  in  the 
second  member  contains  all  the  terms  but  /.  Hence,  if  we 
call  th'^i  sum  of  the  terms  S,  we  have 

S-    a  =  {^  —  l)q  =  ^q  —  lq,    (-'   Sq-^Sz=lq  —  a; 

whence  S  =  — — . 

^-1 


270  ELEMENTARY     ALGEBRA. 

Therefore,  to  obtain  the  sum  of  all  the  terms  or  sum  of  the 
series  of  a  geometrical  progression,  we  have  the 


RULE. 

J.  Multiply  the  last  term  by  the  ratio. 

IJ    Subtract  the  first  term  from  the  product. 

III.  Divide  the  remainder  by  the  ratio  diminished  by  unity 
and  the  quotient  will  be  the  sum  of  the  series. 

1.  Find  the  sum  of  eight  terms  of  the  progression 

2:6:18:  54  :  162  ...  2  X  3'  =  4374. 

q  —  I  2 

2.  Find  the  sum  of  the  progression 

2  :  4  :  8  :  16  :  32. 

q-1  1 

8.  Find  the  sum  of  ten  terms  of  the  progression 

2  :  6  :  18  :  54  :  162  ...  2  X  39  =  39366. 

Ans.  59048 

4.  What  debt  may  be  discharged  in  a  year,  or  twelve 
months  by  paying  |1  the  first  month,  |2  the  second  month, 


168    Give  the  rule  for  finding  the  sum  of  the  series.     What  is  the  fiit4 
step ?     What  the  seaad ?    What  the  third ! 


GEOMETRICAL     PROORB88ION.  271 

^  the  third  month,  and  so  on,  each  succeeding  payment 
being  double  the  last ;  and  what  will  be  the  last  payment  1 

j  Debt,     .     .       $4095. 

(  Last  payment,  |;204b. 

5.  A  gentleman  married  his  daughter  on  New  Year's  day, 
and  gave  her  husband  1^.  towards  her  portion,  and  was  to 
double  it  on  the  first  day  of  every  month  during  the  year : 
what  was  her  portion  1  Ans.  £204  15«. 

6.  A  man  bought  10  bushels  of  wheat  on  the  condition 
that  he  should  pay  1  cent  for  the  first  bushel,  3  for  the  second, 
9  for  the  third,  and  so  on  to  the  last :  what  did  he  pay  for 
the  lust  bushel  and  for  the  ten  bushels  ? 

(  Last  bushel,  |10(>,83. 
"I  Total  cost,     11205,24. 

7.  A  man  plants  4  bushels  of  barley,  which,  at  the  first 
harvest,  produced  32  bushels ;  these  he  also  plants,  which, 
in  like  manner,  produce  8  fold;  he  again  plants  all  hiscn^p, 
and  again  gets  8  fold,  and  so  on  for  16  years  :  what  is  his 
last  crop,  and  what  the  sum  of  the  scries  1 

j  Last,  1407374S83553286U 
^^'     I  Sum,  1C0842843834GG0. 
169.  When    the    progression    is    decreasing,    we    have 
9  <  1  and  /  <  a ;  the  above  formula 

for  the  sum  is  then  written  under  the  form 

in  order  that  the  two  terms  of  the  fraction  may  be  positive. 


1G9.  What  b  the  formula  for  the  sum  of  the  seriee  of  a  decreauDg 
prugre8siou  ! 


272  ELEMENTARY    ALGEBRA. 

1.  Find  the  sum  of  the  terms  of  the  progression 

32  :  16  ;  8  :  4  :  2 

32  -  2  X  4-      oi 

S  =  ^= ^_A=?i^62. 

I  —  q  1  1 

2"  2" 

2.  Find  the  sum  of  the  first  twelve  terms  of  the  progression 

1  / 1  \^^  1 

64  :  16  :  4  :  1  :  —  :...  :  64(—     ,     or 


(t)" 


4 \  4  /    '  65536  * 

64————  V— -    g^fi 
a-Zg_         65536      4  _  65536  65535 

l-^~  £  ~  ^     ~     "^19660& 

4 

170. — Remark.  We  perceive  that  the  principal  difficulty 

consists  in  obtaining  the  numerical  value  of  the  last  term,  a 

tedious  operation,  even  when  the  number  of  terms  is  not 

very  great. 

3.  Find  the  sum  of  6  terms  of  the  progression 

512  :  128  :  32  .  .  . 

Ans.  682J. 

4.  Find  the  sum  of  seven  terms  of  the  progression 

2187  :  729  :  243  ..  . 

Ans.  3279. 

5.  Find  the  sum  of  six  terms  of  the  progression 

972  :  324  :  108  .  .  . 

Ans.  1456 

6.  Find  the  sum  of  8  terms  of  the  progression 

147456  :  36864  :  9216  .  .  . 

Ans.  19660^ 


OXOMBTRICAL     PR 


*^**5^ 


273 


Of  Progressions  having  an  infintte  number  of  lerms, 
171.  Let  there  be  the  decreasing  progression 
a  :  h  :  c  '.  d  '.  e  :  f  :  .  ,  , 
containing  an  indefinite  number  of  terms.     In  the  formulf 

\-q 
substitute  for  /  its  value  a^"-*  (Art.  167),  and  we  have 

1  -q 

which  expresses  the  sum  of  n   terms  of  the  progression. 
This  may  be  put  under  the  form 

s  =  .  "^      "*■ 


l-q       \-q 

Now,  since  the  progression  is  decreasing,  ^  is  a  proper 
fraction ;  and  q*  is  also  a  fraction,  which  diminishes  as  n 
increases.     Therefore,  the  greater  the  number  of  terms  we 

take,  the  more  will    z X  g*    diminish,  and  consequent- 
ly the  more  will  the  entire  sum  of  all  the  terms  approximate 

to  an  equality   with  the  first  part  of  S,  that  is,  to  ^j » 

Finally,  when  n  is  taken  greater  than  any  given  number,  or 
ti  =  infinity,  then    = x  ^   will  be  less  than  any  given 

number,  or  will  become  equal  to  0 ;  and  the  expre&sion  ^j 

will  then  represent  the  true  value  of  the  sum  of  all  the  terms  of 
the  series.     Whence  we  may  conclude,  that  the  expression 

12* 


274  ELEMENTARY     ALGEBRA. 

for  the  sum  of  the  terms  of  a  decreasing  progression^  in  which 
the  number  of  terms  is  infinite,  is 

S  = 


that  is,  equal  to  the  first  term  divided  by  1  minus  the  ratio. 

This  is,  properly  speaking,  the  limit  to  which  the  partial 
eums  approach,  as  we  take  a  greater  number  of  terms  in  the 

progression.     The  difference  between  these  sums  and 

may  be  made  as  small  as  we  please,  but  will  only  become 
nothing  when  the  number  of  terms  is  infinite. 

EXAMPLES. 

1.  Find  the  sum  of 

^•-    3    ^9   ^27^81    '"^"^"^*^- 
We  have  for  the  expression  of  the  sum  of  the  terms 

-  «  1        =^,Ans. 


3 

The  error  committed  by  taking  this  expression  for  tht* 
value  of  the  sum  of  the  n  first  terms,  is  expressed  by 
a 


l-q'^'         2\3/ 
:5;  it 
3  /I  Y_      1  1 

2  \3  /       2  .  3*'~162 


X  g"  = 


First  take  n  =  5 ;  it  becomes 


171.  "When  the  progression  is  decreasioe:  and  the  number  of  terras  ia 
Unite,  what  is  ihe  expression  for  the  vahje  of  the  sum  of  the  series  ? 


OEOMETRIOAL     PROORESSIOir.  275 

WTien  n  =  Q^  we  find 

KlY-JL       i__L 
2  \  3  /   ~162  ^  3  ~486* 

Hence,  we  see,  that  the  error  committed^  by    ^king  ■— 

2 

for  the  sum  of  a  certain  number  of  terms,  is  less  in  proper* 

*Jon  as  this  number  is  greater. 

2.  Again,  take  the  progression 

,11111, 

^   ■    2   •    4    •    8    •   16  •  32        '"" 

We  have        S  =  , = =—  =  2.     Arts, 

3.  What  is  the  sum  of  the  progression 

*'i^'   m'    mo'   mm'    «=<=•.  to  infinity. 

10 

172.  In  the  several  questions  of  geometrical  progression 
there  are  five  numbers  to  be  considered : 

1st.  ITie  first  term, a. 

2d.    The  ratio,       ....          .....  y. 

3d.   The  number  of  terms, n. 

4th.  The  last  term, /. 

5th.  The  sum  of  the  terms, S. 


172    Huw  many  numbers  are  considered  in  geometrical  progiessionl 
What  are  they  I 


276  ELEMENTARY   ALGEBRA. 

173.    We   shall  terminate  this    subject  by  solving  this 
problem. 

To  find  a  mean  proportional  between  any  two  numbers, 
as  m  and  n. 

Denote  the  required  mean  by  x.     We  shall  then  have 
(Art.  156), 

x^  ■=.       m  X  ^, 


and  hence,  x   =  ymXn. 

That  is,  Multiply  the  two  numbers  together^  and  extract  the 

square  root  of  the  product. 

1.  Whcit  is  the  geometrical  mean  between  the  numbers 
2  and  8  1 

Mean  =  -/8  X  2  =  yT6  =  4.  Ans. 

2.  What  is  the  mean  between  4  and  16  ?  Ans.  8. 

3.  What  is  the  mean  between  3  and  27 1  Ans.  9. 

4.  What  is  the  mean  between  2  and  72  ?  A71S.  12. 

5.  What  is  the  mean  between  4  and  64 1  Ans.  16. 

173.  How  do  you  find  a  mean  proportional  between  two  numbers  I 


OF     LOGARITHMS 


277 


CHAPTER  Via 


Of  Logarithms, 


174.  The  nature  and  properties  of  the  logarithms  in  cota- 
mon  use,  will  be  readily  understood,  by  considering  atten- 
tively the  different  powers  of  the  number  10.     They  are, 

10°=  1 

101  ^  10 

102  ^  100 
10'  =  1000 
10*  =  10000 
10*  =  100000 

It  is  plain  that  the  exponents  0,  1,2,  3,  4,  5,  &c.,  form  an 
arithmetical  series  of  which  the  common  diflerence  is  1  j 
and  that  the  numbers  1,  10,  100,  1000,  10000,  100000,  &c., 
form  a  geometrical  progression  of  which  the  common  ratio  is 
10.  The  number  10,  is  called  the  base  of  the  system  of  loga- 
rithms ;  and  the  exponents  0,  1,2, 3,4,  5,&c.,  are  the  loga- 


174i  What  relation  exists  between  the  exponents  1,  2,  8,  <&c.  t     How 

are  the  corresponding  numbers  10,  100,  1000?  What  is  the  common 
difitTcrice  of  the  exjxments !  What  is  tlie  common  ratio  of  the  corrt'S 
|><>r»iliiiij  niunlj^Ts  ?  What  is  the  ba.«e  of  tlie  common  sjsttMn  of  lopa 
rithms  I  What  are  the  exponents  f  Of  what  number  is  the  exponent 
1  tlie  luf^aritiim  t    Tli«  exponent  2  t    The  exfMjneiit  8  t 


278  ELEMENTARY     ALGEBRA, 

rithms  of  the  numbers  which  are  produced  by  raising  10  tc 
^he  powers  denoted  by  those  exponents. 

175.  If  we  denote  the  logarithm  of  any  number  by  w, 
then  the  number  itself  will  be  the  mth  power  of  10  :  that  is, 
if  we  represent  the  corresponding  number  by  M^ 

10°^  =  M. 
Thus,  if  we  make  m  =  0,  if  will  be  equal  to  1 ;  if  »i  =  1, 
M  will  be  equal  to  10,  &c.     Hence, 

The  logarithm  of  a  number  is  the  exponent  of  the  power  to 
which  it  is  necessary  to  raise  the  base  of  the  system  in  order 
to  produce  the  number. 

176.  Letting,  as  before,  10  denote  the  base  of  the  system 
jf  logarithms,  m  any  exponent,  and  M  the  corresponding 
number :  we  shall  then  have, 

lO'"  =  M 
in  which  m  is  the  logarithm  of  M. 

If  we  take  a  second  exponent  n,  and  let  iV  denote  the 
corresponding  number,  we  shall  have, 

in  which  n  is  the  logarithm  of  iV. 

If  now,  we  multiply  the  first  of  these  equations  by  the 
second,  member  by  member,  we  have 

10""  X  lO"*  =  10'"+'»  =  if  X  iV; 
bat  since  10  is  the  base  of  the  system,  m  -{•  n  is  the  loga- 
rithm M  X  N  \  hence, 

175i  If  \7e  denote  the  base  of  a  system  by  10,  and  the  exponent  by 
m,  "what  •will  represent  the  corresponding  number  ?  What  is  the  loga- 
rithm of  a  number  t 

176i  To  what  is  the  sum  of  the  logarithms  of  any  two  numbers  equal  \ 
To  what  then,  -will  the  addition  of  logarithms  correspou'l  f 


or     LOGARITHMS. 


279 


The  sum  of  the  logarithms  of  any  two  numbers  is  equal  to 
the  logarithm  of  their  product. 

Therefore,  the  addition  of  logarithms  corresponds  to  the 
multiplication  of  their  numbers. 

177.  If  we  divide  the  equations  by  each  other,  member 
by  member,  we  have, 

—  lO"""""  — • 

10°  ~  ~N' 

but  since  10  is  the  base  of  the  system,  m  —  n  is  the  loga 
rithm  of  -77  ;  hence, 

IV 

If  one  number  he  divided  by  another^  the  logarithm  of  the 
quotient  will  be  equal  to  the  logarithm  of  the  dividend  dimi- 
nished by  that  of  the  divisor. 

Therefore,  t/ie  subtraction  of  logarithms  corresponds  to  the 
division  of  their  nutnbers. 

178.  Let  us  examme  further  the  equations 

100  =  1 

101  =  10 

102  ^  100 

103  ^  1000 
&c.      &c. 

It  is  plain  that  the  logarithm  of  1  is  0,  and  that  the  loga- 
lithms  of  all  the  numbers  between  I  and  10,  are  greater 
than  0  and  less  than  1.  They  are  generally  expressed  by 
decimal  fractions  ;  thus, 

log  2  =  0.3010G0. 

177i  If  one  number  be  divided  bj  another,  what  will  the  logarithm 
of  tlie  quotient  be  equal  to !  To  what  tlien  will  the  subtraction  of 
logaritluns  correspond  7 

178.  What  is  the  logarithm  of  1  ?  Between  what  liraita  are  the  loga 
ritlims  of  all  numbers  between  1  and  10  ?  How  are  thuy  genoraUy 
ffxpreiaedt 


280  KLEMENTARY     ALGEBRA. 

The  logarithms  of  all  the  numbers  greater  than  10  and 
less  than  100,  are  greater  than  1  and  less  than  2,  and  are 
generally  expressed  by  1  and  a  decimal  fraction  :  thus, 

log  50  =  1.698970. 

The  part  of  the  logarithm  which  stands  on  the  left  of  the 
decimal  point,  is  called  the  characteristic  of  the  logarithm, 
rhe  characteristic  is  always  07ie  less  than  the  number  of 
places  of  figures  in  the  number  whose  logarithm  is  taken. 

Thus,  in  the  first  case,  for  numbers  between  1  and  10, 
there  is  but  one  place  of  figures,  and  the  characteristic  is  0. 
Tor  numbers  between  10  and  100,  there  are  two  places  of 
figures,  and  the  characteristic  is  1  ;  and  similarly  for  other 
numbers. 

Tahh  of  Logarithms, 

179.  A  table  of  logarithms  is  a  table  in  which  are  writ- 
ten the  logarithms  of  all  numbers  between  1  and  some 
other  given  number.  A  table  showing  the  logarithms  of 
the  numbers  between  1  and  100  is  annexed.  The  numbers 
are  written  in  the  column  designated  by  the  letter  N,  and 
the  logarithms  in  the  columns  designated  by  Log. 

How  is  it  with  the  logarithms  of  numbers  between  10  and  100 
What  is  that  part  of  the  logarithm  called  which  stands  at  the  left  of 
the  characteristic  ?     What  is  the  value  of  the  characteristic  ? 

179i  What  is  a  table  of  logarithms  ?  Explain  the  manner  of  finding 
the  logarithms  of  niunbers  between  1  and  IJOI 


OF      LOGARITHMS, 

TABLE. 


us\ 


N. 

L«»^. 

1 

0.000000 

2 

0.301030 

3 

0.477121 

4 

0.(J020(>0 

5 

0,09.SJ>70 

G 

0.778151 

7 

0.84501)8 

8 

0.1)030i>C 

l> 

0.954243 

10 

1.000000 

h 

1.041393 

12 

1.079181 

13 

1.113943 

14 

1.14()128 

15 

1.170091 

!G 

1.204120 

17 

1.230449 

lb 

1.255273 

19 

1.278754 

20 

1.301030 

21 

1.322219 

22 

1.342423 

23 

1.301728 

24 

1.380211 

25 

1.397940 

.og. 


2(3j  1.414973 
27 1 1.431 304 
2811.447158 
29 1 1.402398 
3011.477121 


31 
32 
33 
34 
35 


1.491302 
1.505150 
1  518514 
1.531479 
1.544068 


30  1 

38' 1 
,39,1 

l|40  1 

Uljl 
42  1 
4311 
4411 

4()  1 


47 

48 
49 
50 


.550303 
.508202 
.579784 
.591005 
.002060 
.612784 
.623249 
.633408 
.043453 
.653213 
.662758 
.672098 
.681241 
.690 1 90 
.698970 


N. 

Log. 

N. 

51 

1.707570 

76 

52 

1.716003 

77 

53 

1.724276 

78 

54 

1.732394 

79 

55 

1.740363 

80 

56 

1.748188 

81 

57 

1.755875 

82 

58 

1763428 

83 

59 

1.770852 

84 

60 

1.778151 

85 

61 

1.765330 

86 

62 

1.792392 

87 

63 

1.799341 

88 

64 

1.800180 

89 

65 

1.812913 

90 
91 

ijij 

1.819544 

67 

1.826075 

92 

68 

1.832509 

93 

69 

1.838849 

94 

70 

1.845098 

95 

71 

1.851258 

66 

72 

1.857333 

97 

73 

1.863323 

98 

74 

1.869232 

99 

75 

1.875061 

100 

Log. 
1.880814 
1.886491 
1 .892095 
1.897627 
1.903090 


1.908485 
1.913814 
1.919078 
1.924279 
1.929419 
r934498 
1.939519 
1.944483 
1.949390 
1.954243 


1.959041 
1.963788 
1.968483 
1.973128 
1.977724 


1.982271 
1.986772 
1.991226 
1.995635 
2.000000 


EXAMPLES. 


1.  Let  it  be  required  to  multiply  8  by  9,  by  means  of  loga- 
rithms. We  have  seen,  Art.  176,  that  the  sum  of  the  loga- 
rithrns  is  equal  to  the  logarithm  of  the  product.  Therefore, 
find  the  logarithm  of  8  from  the  table,  which  is  0.903090. 
and  then  the  logarithm  of  9,  which  is  0.954243  ;  and  their 
«um,  which  is  1.857333,  will  be  the  logarithm  of  the  product. 

18 


282 


ELEMENTART     ALGEBRA, 


In  searching  along  in  the  table,  we  find  that  72  stands  oppo 
site  this  logarithm  :  hence,  72  is  the  product  of  8  by  9, 


2.  What  is  the  product  of  7  by  12? 
Logarithm  of  7  is,     . 
Logarithm  of  12  is,    . 

Logarithm  of  their  product, 
and  the  number  corresponding  is  84. 

3.  What  is  the  product  of  9  by  11  ? 
Logarithm  of  9  is,      . 
Logarithm  of  11  is,   . 

Logarithm  of  their  proauct, 
and  the  corresponding  number  is  99. 


0.845098 
1.079181 

1.924279 


0.954243 
1.041393 

1.995636 


4.  Let  it  be  required  to  divide  84  by  3.  We  have  seen 
in  Article  177,  that  the  subtraction  of  Logarithms  corres- 
ponds to  the  division  of  their  numbers.  Hence,  if  we  find 
the  logarithm  of  84,  and  then  subtract  from  it  the  logarithm 
of  3,  the  remainder  will  be  the  logarithm  of  the  quotient. 


The  logarithm  of  84  is,      .         . 
The  logarithm  of  3  is, 

.  1.924279 
.     0.477121 

Their  difference  is,      . 
and  the  number  corresponding  is  28. 

5.  What  is  the  product  of  6  by  7 1 
Logarithm  of  6  is,     . 
Logarithm  of  7  is,     . 

.     1.447158 

.  0.778151 
.     0.845098 

Their  sum  is. 


1.623249 


&ad  Uie  corresponding  number  of  the  table,  42. 


SUPPLEMENT. 

XXAMPLES     IN     ADDITION     AND     SUBTRACTION, 

1.  What  is  the  sum  of 

ax^  +  b3f  and   cx°  -h  dj^, 

2.  What  is  the  sum  of 

aj;°   and   bx''  —  cx°  —  c?x°. 

8.  What  is  the  sum  of 

10a*  4- 3a*   and   6a*  —  a*  —  5a' 

4.  What  is  the  sum  of 

5a^  —  7a^    and    lla^  -f-  a*. 

5.  What  is  the  sum  of 

a'i"  —  9a"  +  5a"6"   and   GaT'  +  10a''6'". 

6.  What  is  the  sum  of 
ba^b^-\-7ab^c-Sa"'b^-l2ab^-c    and   Qa^^-9a^b^+b'-8a''bf 

-362. 

7.  What  is  the  sum  of 

5a*6  +  Sa^b^c  —  lab  —  6a*b    and    2a26-c  +  17a6  -f-  '^a*b  - 
8a262c  —  10a6. 

8.  What  is  the  sum  of 

So"*'  +  Sa^i"-'  —  3a^  —  3m"6P    and    4r72a36'»->  —  a  -f  lOa* 
4-  a-'fiP  +  a  H-  SaH^  —  2r/^a^"^\ 

9.  What  is  the  sum  of 

9a»63c*  _  76  -I-  186  —  Da^i"  +C  —  3t/«  and  3o"6™  —  ha^h'^e* 
+  3c'  -  5rf5. 
10    From    -  Oa^a;*  —  13 -|- 2a63x  -  46"'cj;2 
take      36"c^»  —  9a''x^  —  0  4-  2a63^. 


284       ELEMENTARY     ALGEBRA SUPPLEMENT. 


11.  From 
t<ake 

12.  From 
take 

13.  From 
take 

14.  From 
take 

15.  From 
take 


-  IbaW  +  3a*  -  Sa^  -  lcd\ 
^aH'  4-  Gai"*  —  d^  +  18a*^° 
7a*i»  4-  ^^  —  Sai"^  +  9a'"^>2. 
1265^6  _  lOaS^e  -  Sa"^''  +  6cm* 
Ga'^Z*"  —  6ac^  +  IGaSfis  +  126^(^6. 
8^353^5  _  i2arb  +  Gaa;4  +  Sac?"* 
SatZ"^  —  8a353c5  —  12a'°6  +  6aa:*. 
12a"Z»°  —  9a.r5  —  4a6  +  6^252  —a 
3a  —  6a262  +  12a'"6°  —  Qaa;*  -f  5a6. 


EXAMPLES    IN    MULTIPLICATION. 

1.  What  is  the  simplest  form  of  the  product  of 

a"  X  a\ 

2.  What  is  the  simplest  form  of  the  product  of 

2a3  X  7a9  X  -  3a*. 

3.  What  is  the  simplest  form  of  the  product  of 

a^b  X  a'd  X  10a  X  6a2  x  —  1. 

4.  What  is  the  simplest  form  of  the  product  of 

_  ar-i  X  —  3a^2  xfX  Sa^'i+'^car. 

5.  What  is  the  simplest  form  of  the  product  of 

ba^b^  X  lOa^^c  X  -  3a'^. 

6.  What  is  the  simplest  form  of  the  product  of 

-  7a56V  X  Sa^bU  X  Ib^c  X  -  1. 

7.  What  is  the  simplest  form  of  the  product  of 

arb\q  X  a°b'c'i  X  a'^b  X  —  a. 
S,  What  is  the  simplest  form  of  the  product  of 

(a2  —  Sab  —  5^2)  x  4.a'^b. 
9.  What  is  the  simplest  form  of  the  product  of 


EZAMPLX8     IN     MULTIPLICATION.  2S5 

10.  What  is  the  simplest  form  of  the  product  of 

{IhH  +  2/3  -  3aAa/2  ^  7)  X  -  8A*/^ 

11.  What  is  the  simplest  form  of  the  product  of 

{a?b^  -  cbH^f^  3c-)  X  -  2k2c/. 

12.  What  is  the  simplest  form  of  the  product  of 

13.  What  is  the  simplest  form  of  the  product  of 

(3A:2  _  5;^,/  ^  2/-^)  X  (F  -  Ikl). 

14.  What  is  the  simplest  form  of  the  product  of 

(C/2  -  \lfl  +  3/2)  X  (P  +  4/*/). 

15.  What  is  the  simplest  form  of  the  product  of 

(4a2  _  lOoar  +  3^:2)  x  (Sa^  -  2tt2jr). 

16.  What  is  the  simplest  form  of  the  product  of 

(a2-|-a*  +  a6)  X  (a?- 1). 

17.  What  is  the  simplest  form  of  the  product  of 
(a*  -  2a^  -f  4a262  -  SaP  +  IGM)  X  (a  -f-  26). 

18    What  is  the  simplest  form  of  the  product  of 
(2a^2:2  _  3i4y2)  X  (2a*a:2  -f-  36^^). 

19.  What  is  the  simplest  form  of  the  product  of 

(7a3  -  5a26  -f  Gab^  —  2b^)  X  (3a*  -  4a^6  -f  lGa2/>*).     .' 

20.  What  is  the  simplest  form  of  the  product  of 
(a«  -  3a«62  -f  baH*)  X  (7a*  -  ^a^^  +  6*). 

EXAMPLES    IN    DIVISION.. 

1    Divide  a"  by  a\ 

2.  Divide  a"  by  a^". 

3.  Divide  Sa^*  by  2a*. 

4.  Divide  ca^*  by  ia*. 


286       ELEMENTARY     ALGEBR  A S  UPPLBMBNT. 

5.  Divide  6{a  +  by  by  3(a  +  by. 

6.  Divide  (a  +  xf  X  (a  +  yf  by  (a  +  x)  X  {a -{-  y)\ 

7.  Divide  QaW  -  15ay+  21a%x,  by  Sa^. 

8.  Divide  bc^  —  c^x  by  b  —  x. 

9.  Divide  a^  -f-  a?b  —  ai^  —  6^  by  a  —b. 

10.  Divide  Sa^  -f  l6a*6  -  SSa^fis  -f  \4.a%^  by  a2  +  7a5. 

11.  Divide  a?  -  Qa%^  +  Ma^b^  -  12^459  by  a^  -  2a263. 

12.  Divide  a*  —  2a2Z»2  -|_  ^4  \^y  ^2  _  52, 

13.  Divide   -  a%^  +  ISa^i^s  -  4Sa^*b^  —  20ai^^»'' 
by  I0a%^  —  a%. 

14.  Divide  a«  —  160"  by  a^  —  2^2. 

15.  Divide  2a*  -  ISa^i  +  ^laH^  -  38a63  4-  2^4 
by  2a2  _  3a6  +  462. 

1(5.  Divide  4c*  -  9b'='^c^  +  GS^c  —  b^  by  2c2  —  36c  -f  6^, 

17.  Divide  —  1  +  a^fi^  by   —  1  -f  an, 

18.  Divide  a^  -\-  2a^2^  +  2^  by  a^  —  az  -{-  z'^. 

19.  Divide  1  _  6^2  +  27^*  by  ^  +  2^  +  3^2. 

20.  Divide  a^  —  IGa^x^  -f  64^^  by  a2  -  4aa;  +  4:x\ 

21.  Divide  a^c^  —  Sa'^cd^  +  3ac2c^3  _  cW  +  a2c2i;2  _  ^^^^ 
by  a2c?2  —  2acd^  +  c'^d^  4  ac2fl?. 

EXAMPLES    IN   REDUCTION    OF    FRACTIONS. 

1.  Reduce  to  its  simplest  terms  the  fraction 

18ac/— 66(/c/— 2ac? 
3adf 

2.  Reduce  to  its  simplest  terms  the  fraction 

8a2  -  6a6  +  4c  -f  1 
—  2a 


1XAMPLE8    IN    REDUCTION    OF    FRACTIONS.    287 

3.  Reduce  to  its  simplest  terms  the  fraction 

4a26y^. 

4.  Reduce  to  its  simplest  terms  the  fraction 

ab  —  ac 
b-c' 


ex 

5.  Reduce  a  —  b  -\ to  the  form  of  a  fraction. 

X  —  a 

b  —  y 

6.  Reduce   x to  the  form  of  a  fraction. 

ax  —  c 


y  —  a, 
7.  Reduce   a  -\-  b  -\-  — i—   to  ►he  form  of  a  fraction. 
a 


8.  Reduce   x  —  ab p  the  form  of  a  fraction. 


9.  Reduce   a lo  the  form  of  a  fraction. 

X  —  y 

10.  Reduce   6a/^x-\-9a/ —  to  the  form  of  a  fractioa 


X  ~~  ax 

II.  Reduce   5acx  —  t to  the  form  of  a  fraction, 

fac 

12»  Reduce  to  an  en  iire,  or  mixed  quantity,  the  fraction 

2a2 
'3.  Reduce  U        entire,  or  mixed  quantity,  the  fraction 


288  ELEMENTARY  ALGEBRA SUPPLEMENT. 

14.  Reduce  to  an  entire,  or  mixed  quantity,  the  fraction 

a^  —  2o?x  -f-  ab  +  ax^  —  bz 
a  —  X 

15.  Reduce  the  following  fractions  to  a  common  denomt 
nator :  viz. 

a  —  X      h  —  c    4ax  —  c 


a  -\-  x^       /         a  —  X 

16.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

a  a  —  X        ^  c 

and . 


Zax  -  V       36 

17.  Reduce  the  following  fractions  to  a  common  denomi 
nator :  viz. 

4.af—  X       a  —  X        _     5ac 

■^ , and   . 

la  —  c  c  y 

18.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

4:ax  -h  6  Sac  —  / 

8ac  —  /  4,ax 

19.  Reduce  the  following  fractions  to  a  common  denomi- 
nator:  viz. 

a  -\-  x     a  —  b         ,c —  d 

,     — ; and    . 

a  —  X      a  -\-  X  a 

20.  Reduce  the  following  fractions  to  a  common  denoiiu 
Dator :  viz. 

a  -f  ft  —  c     c  -f/        ,    X  ~  a 

, and    — - — . 

X  —a  c  X  -\-  a 


XAMPLEB    IN    ADDITION    AND    SUBTRACTION.    CBS 


U>DITION,    SUBTRACTION,    MULTIPLICATION,    AND    DIVISION    OF 
FRACTIONS. 

1.  What  is  the  sum  of ,    and  c? 

X  —  a     a  —  X 


Ct  ~~  X      C  ~^   CL 

2.  What  is  the  sum  of  — r — , and  y  1 

b         a  —  c 

3.  What  IS  the  sum  of  — ,    — -—-, ? 

0  a  oc  — J 

4.  What  is  the  sum  of  ^^  ~  ,    ^^  ~^,    5ay  ?• 

c  -f-  a        —  a;  ^ 

5.  What  is  the  sum  of   3a  H ,  2a  —  r- 

X  —  a^  o  4  ^ 

6.  From  8a  +  —  take . 

0  a  —  X 

^    _,  Sx  —  ax  ^  .      „        5aa; 

7.  From  j take  3 . 

0  c 

^    _  ax  -{•  Say      .      2ax  —  of 

8.  From  — — — ^  take ^. 

acx  —  ay  ay 

-n.    -r^  fl^  4"  ^      ,      «  6aar  —  y 

9.  r  rom  ay take  3ay  H — ^ 

az  9 

10.  From  intake  7a6^-^r^. 

Sx  8a  —  X 

or  

11.  Multiply  7a +  —  by    --— . 

.«■».,.,         5a      ,      „  5aar  —  a:* 

12.  Multiply    by  3ay 

^  "^    X  —  a     ''  X 

13 


290   ELEMENTARY  AIGEBR  A S  UPPLEMENT. 


,„-^,.-      9a  —  X  ^      ^         (jax  —  x^ 
13    Multiply  -— - —  Lj  2a  + 


^a  +  X 


9a 


14   Multiply  Sax  -\-  ^^L^   by  5  + 


15.   Multiply   Qa  -\ by 


-b 


a"  -  62' 


16.   Divide  Mc  —  2adc  —  f -{- —  by  2a. 

a 


a    ,    /c? 


3c 


17.   Divide  —  +  ^  _  3ac  +  7   by  — , 
d         2c  "^     d 


i8.   Divide  3a2  - 

7a6         21ac        562    ^    Qsbc        3^ 
2             4            2      '       8            2 

by    3a  -  56  +  ^. 

19,    Divide   2/2  - 

55/^       29/a;       21/^2       15^^       3.% 
12     '      9     '      8            4     "^   3 

.     2/        3A^ 

20.   Divide ^  +  -g^ -g-  +  -^   +  25^2 

^y  --^  +  52/. 


EXAMPLES    m    EQUATIONS    OP    THE    FIRST    DEGREE. 


1,  Given   ■> 


4x 


X  —  y 


4-  7a;  =  41 


to  find  X  and  y. 


BQUATIONS     OF     THE     FIRST     DEGREE.         291 

2.  Given 


"~4        ^""5""    ^^    y   to  find  ar  and  y. 


-y+  4iy    =12i 


8.  Given  ax  +  -^-  +  l(x  -/)  =  g  U)  find  x. 


X 

c  —  a 


4.  Given  i^  +  5:^  -  ^^^  =  rf,  to  find  «. 
4  5c 


5.  Given 


6.  Given 


7.  Given 


Sy  —X   ^  2x  —  y  __ 
4 
8-22? 


6 

6a;  — y + 


+  — =1=    5 


43J 


.  to  find  X  and  y. 


3j!  — 8   ,  y-6  ,    __ 


+  ^  +  y  =  18A 


8,_3-^=79 


to  find 
a;  and  y. 


8.  Given  "-II^  -  i^^  +  :r  =  J,  to  find  x. 

9.  Given  ^1:=^? -?^^1^ +  « -<i=/  to  find  ,. 

6  c 

10.  Given    {^-"y  +  .I^}    »<>  ^-^^  ^ -^  V" 

U.  Given  i=--  +  ^-  +  4  (.r  -  3)  =  68,  to  find  * 


292   ELEMENTARY  ALGEBRA SUPPLEMPaT. 


12.  Given 


13.  Given 


14.  Given 


>    to  find  X,  y  *'-^4  z. 


i  a:  +  2y  -h  32 
\  z  —  y-\-zz=z 
(  3a;  +  6y  +  2 


=  14 

2         J-  to  find  x^  V  and  z, 

=  18 


TXT 


t:z3^y^zl^s 


>    to  find  x^  y  and  z. 


find 
and  y 


15.  Given     {  f^^Vy"'^^^^  =^^+431.  |  ^o  find  .  and  y. 

16.  Given     i  (-+5)(y+T)=(.+  l)(y-9)  +  n2  )    to 

( 2a;4-10=3yH-l  ja;ar 

17.  Given     J   "^  —    y  [  %o  find  a:  and  y, 

18.  Given     |^^  ^  ^^^  ~  ^^  j-  to  find  x  and  y. 

» 9.  Given     -j  j  _j_  y        3a  -|-  .r  f   to  find  x  and  y. 
iax  +  2by  =  d      ) 


20.  Given 


(  6ca;  =  cy  ~2b 


to  find  ar  and  y. 


'^l.  Given     ^ 


—  26/*'  r  t^  ^^^  ^  ^^^  y« 


BQUATI0N8     OF     THK     FIRST     DEORKB. 


293 


22.  Given    )x-^y  —  z=  18J  [  to  find  ar,  y  and  s. 
(a:  —  y  -h  z 


13J 


(Sx-\-5y  =  161  J 
23    Given    )lx-{-2z=  209  V  to  find  j,  y  and  «. 
(  2y  -f    2  =  89    ) 


a-        y 

24.  Given   ^ 

X           z 

L  y          ^           J 

'  to  find  ar,  y  and 

r  oa:  4-  &y  =  c  ^ 

25.  Given    <^dx  -\-  ey  =/  ^  to  find  ar,  y  and  ar. 

[ffy~^hz  =  l) 

EXAMPLES    IN   EQUATIONS    OF   THE   SECOND   DEOREB 

1.  Given  x^  —  5 Jar  =  18  to  find  x, 

2.  Given  3ar2  —  2ar  =  65  to  find  ar. 

3.  Given  622ar  =  15ar2  +  6334  to  find  x. 

4.  Given  11  Jar  -  Sjx^  =  _  41},  to  find  ar. 

5.  Given  9jar2  _  901^;  ^  195  =  0,  to  find  ar. 

6.  Given  20748  -  1616ar  +  21ar2  _  q,  to  find  x. 

7.  Given  9}ar2  —  90Jx  +  195  =  0,  to  find  ar. 

S,  Given     "^^  -f  -^'^  +  4728  =  0,  to  find  x. 
5  C5 


294   ELEMENTARY  ALGEBRA SUPPLEMENT. 

9.  Given  x^  —  8a;  =  14   to  find  x. 

10.  Given  Zx^-\-x  =  l   to  find  a;. 

11.  Given  118a;  —  2ja;2  =  20   to  find  x, 

12.  Given  6a:  —  30  =  Zx^   to  find  x, 

13.  Given  8a;2  —  7a;  +  34  =  0   to  find  a;. 

14.  Given  4a;2  -  9a;  =  5a;2  —  255}  -  8a;   to  find  «. 

^r,    n-  oA     ,    3^'   ,   21a; -27782       ^__,       _  . 

15.  Given    80a;  +  — -  H — =  1859J  —  3x* 

16.  Given     —-  =  — — -—  +  1    to  find  x, 

6a;       117  — 2a; 

,^    ^.  25a; +180         40a;  3    ^    .    , 

17.  Given    -— —  = r  -  —   to  find  x, 

10a;  —  81       5a;  —  8       5 

^^    ^.  18  + a;  20a; +  9  65         ^     .    , 

18.  Given     g^— ^  =  ^^-^^  -  ^^-^    to  find  *. 

19.  Given     a;^  —  7a;  +  3^  =  0   to  find  x, 

20.  Given    4a;2  —  9a;  =  5a;2  —  255f  —  8a;  to  find  x, 

X  7 

21.  Given     — ,   ^^   = —  to  find  a;. 

a?  +  60        3a;  —  5 

40  27 

22.  Given    — ^  +  IL  =  13   to  find  x. 

a;  —  5        X 

23.  Given    -?l_-6=?^  to  find  ». 

a;  +  2  3a; 

^i.  Given =  — -— -  —  5   to  find  a?. 

a;  +  3       a;  +  10 


PROMISCUOUS  PROBLEMST 

eiVINe  RISE  TO 

EQUATIONS  OF  THE   P^IRST  DEGREE. 

1.  A  person  expended  30  cents  for  apples  and  pears, 
giving  one  cent  for  four  apples,  and  one  cent  for  five  pears : 
he  then  sold,  at  the  prices  he  gave,  half  his  apples  and  one- 
third  his  pears,  for  13  cents.  How  many  did  he  buy  of 
each? 

2.  A  tailor  cut  10  yards  from  each  of  three  equal  pieces 
of  cloth,  and  17  yards  from  another  of  the  same  length, 
and  found  that  the  four  remnants  were  together  equal  to  142 
yards.     How  many  yards  in  each  piece  1 

3.  A  fortress  is  garrisoned  by  2G00  men,  consisting  of 
infiiutry,  artillery,  and  cavalry.  Now,  there  are  nine  times 
as  many  infantry,  and  three  times  as  many  artillery  soldiers, 
as  there  are  cavalry.     How  many  are  there  of  each  corps'? 

4.  All  the  joumeyings  of  an  individual  amounted  to  2970 
miles.  Of  these  he  travelled  3J  times  as  many  by  water 
as  on  horseback,  and  2}  times  as  many  on  foot  as  by  water. 
How  many  miles  did  he  travel  in  each  way  1 

5.  A  sum  of  money  was  divided  between  two  persons, 
A  and  B.  A's  share  was  to  B's  in  the  proportion  of  5  to  3, 
and  exceeded  five- ninths  of  the  entire  sum  by  50.  What 
was  the  share  of  each  1 

6.  There  are  52  pieces  of  money  in  each  of  two  begs,  out 
of  which  A  and  B  help  thomsolves.    A  takes  twice  as  much 


296   ELEMENTARY  ALGEBRA SUPPLEMENT. 

as  B  leaves,  and  B  takes  seven  times  as  much  as  A  leaves. 
How  much  does  each  take  ? 

7.  Two  persons,  A  and  B,  agree  to  purchase  a  house  to- 
gether, worth  $1200.  Says  A  to  B,  give  me  two-thirds  of 
your  money  and  I  can  purchase  it  alone ;  but,  says  B  to  A, 
if  you  give  me  three-fourths  of  your  money  I  shall  be  able 
\r  purchase  it  alone.     How  much  had  each  ? 

8.  A  father  directs  that  $1170  shall  be  divided  among 
nis  three  sons,  in  proportion  to  their  ages.  The  oldest  is 
twice  as  old  as  the  youngest,  and  the  second  is  one-third  older 
than  the  youngest.     How  much  was  each  to  receive  ? 

9.  Three  regiments  are  to  furnish  594  men,  and  each  to 
furnish  in  proportion  to  its  strength.  Now,  the  strength  of 
the  first  is  to  the  second  as  3  to  5  ;  and  that  of  the  second 
to  the  third  as  8  to  7.     How  many  must  each  furnish  ? 

10.  A  grocer  finds  that  if  he  mixes  sherry  and  brandy  in 
the  proportion  of  2  to  1,  the  mixture  will  be  worth  785.  per 
dozen  ;  but  if  he  mixes  them  in  the  proportion  of  7  to  2,  he 
can  get  79s.  a  dozen.  What  is  the  price  of  each  liquor  per 
dozen  ? 

11.  A  person  bought  7  books,  the  prices  of  which  were  in 
arithmetical  progression,  (in  shillings).  The  price  of  the  one 
next  above  the  cheapest,  was  8  shillings,  and  the  price  of  the 
dearest,  23  shillings.     What  was  the  price  of  each  book  ] 

12.  A  number  consists  of  three  digits,  which  are  in  arith- 
metical proportion.  If  the  number  be  divided  by  the  sum 
of  the  digits,  the  quotient  will  be  26 ;  but  if  198  be  added 
to  it,  the  order  of  the  digits  will  be  inverted. 

13.  A  person  has  three  horses,  and  a  saddle  which  is  worth 
1220.     If  the  saddle  be  put  on  the  back  of  the  first  horse,  it 


EQUATIONS     OF     THE     FIRST     DEORSB.  297 

will  make  his  value  equal  to  that  of  the  second  and  third ; 
if  it  be  put  on  the  back  of  the  second,  it  will  make  his  value 
double  that  of  the  first  and  third ;  if  it  be  put  on  the  back 
of  the  third,  it  will  make  his  value  triple  that  of  the  first 
and  second.     What  is  the  value  of  each  horse  1 

14.  The  crew  of  a  ship  consisted  of  her  complement  of 
sailors,  and  a  number  of  soldiers.  There  were  22  sailors  t<? 
every  three  guns,  and  10  over ;  also,  the  whole  number  of 
hands  was  five  times  the  number  of  soldiers  and  guns  to- 
gether. But  after  an  engagement,  in  which  the  slain  were 
one-fourth  of  the  survivors,  there  wanted  5  men  to  make  13 
men  to  every  two  guns.  Required,  the  number  of  guns, 
soldiers,  and  sailors. 

15.  Three  persons  have  |96,  which  they  wish  to  divide 
equally  between  them.  In  order  to  do  this,  A,  who  has  the 
most,  gives  to  B  and  C  as  much  as  they  have  already :  then 
B  divides  with  A  and  C  in  the  same  manner,  that  is,  by 
giving  to  each  as  much  as  he  had  after  A  had  divided  with 
them :  C  then  makes  a  division  with  A  and  B,  when  it  is 
found  that  they  all  have  equal  sums.  How  much  had  each 
at  first  1 

16.  To  divide  the  number  a  into  three  such  parts,  that 
the  first  shall  be  to  the  second  as  m  to  n,  and  the  second  to 
the  third  as  p  to  q. 

17.  Five  heirs,  A,  B,  C,  D  and  E,  are  to  divide  an  inher- 
itance of  15600.  B  is  to  receive  twice  as  much  as  A,  and 
1200  more ;  C  three  times  as  much  as  A,  less  $400 ;  D  the 
half  of  what  B  and  C  receive  together,  and  150  more  ;  and 
E  the  fourth  part  of  what  the  four  others  get,  plus  $475. 
How  much  did  each  receive  ] 

18.  A  person  has  four  casks,  the  second  of  which  being 

13* 


298       ELEMENTARY     ALQEBRA SUPPLEMENT. 

filled  from  the  first,  leaves  the  first  four-sevenths  full.  The 
third  being  filled  from  the  second,  leaves  it  one-fourth  full, 
and  when  the  third  is  emptied  into  the  fourth,  it  is  found  to 
fill  only  nine-sixteenths  of  it.  But  the  first  will  fill  the  third 
and  fourth,  and  leave  15  quarts  remaining.  How  many- 
quarts  does  each  hold  *? 

19.  A  courier  having  started  from  a  place,  is  pursued  by 
\a  second  after  the  lapse  of  10  days.     The  first  travels  4 

miles  a  day,  the  other  9.  How  many  days  before  the 
second  will  overtake  the  first  ? 

20.  If  the  first  courier  had  left  n  days  before  the  other, 
and  made  a  miles  a  day,  and  the  second  courier  had  travelled 
b  miles,  how  many  days  before  the  second  would  have  over- 
taken the  first  ? 

21.  A  courier  goes  31 J  miles  every  five  hours,  and  is  fol- 
lowed by  another  after  he  had  been  gone  eight  hours.  The 
second  travels  22J  miles  every  three  hours.  How  many 
hours  before  he  will  overtake  the  first  1 

22.  Two  places  are  eighty  miles  apart,  and  a  person  leaves 
one  of  them  and  travels  towards  the  other,  at  the  rate  of  3| 
miles  per  hour.  Eight  hours  after,  a  person  departs  from  the 
second  place,  and  travels  at  the  rate  of  5J  miles  per  hour. 
How  long  before  they  will  meet  each  other  1 

23.  Three  masons.  A,  B  and  C,  are  to  build  a  wall.  A 
and  B  together  can  do  it  in  12  days  ;  B  and  C  in  20  days  ; 
and  A  and  C  in  15  days.  In  what  time  can  each  do  it  alone, 
and  in  what  time  can  they  all  do  it  if  they  work  together  ? 

24.  A  laborer  can  do  a  certain  work  expressed  by  a,  in  a 
time  expressed  by  6  ;  a  second  laborer,  the  w^ork  c  in  n  time 
c?;  a  third,  the  woik  e  in  a  time/     It  is  required  to  find  the 


EQUATIONS  OF  THE  FIRST  DKOREK, 


299 


le  it  would  take  the  throe  lahorcrs,  working  together,  to 
•m  the  work  </. 

25.  Required  to  find  three  numbers  with  the  following 
editions.     If  G  be  added  to  the  1st  and  2d,  the  sums  are 

one  another  as  2  to  3.     If  5  be  added  to  the  1st  and  8d, 
le  sums  are  as  7  to  11 ;  but,  if  3G  be  subtracted  from  the 
and  3d,  the  remainders  will  be  as  G  to  7. 

26.  The  sum  of  $500  was  put  out  at  interest,  in  two 
jparate  sums,  the  smaller  sum  at  two  per  cent,  more  than 
ie  other.  The  interest  of  the  larger  sum  was  afterwards 
icreased,  and  that  of  the  smaller  diminished  by  one  per 
mt.  By  this,  the  interest  of  the  whole  was  augmented  one- 
>urth.  But  if  the  interest  of  the  greater  sum  had  been  so 
icreased,  without  any  diminution  of  the  less,  the  interest  of 
le  whole  would  have  been  increased  one-third.  What  were 
le  sums,  and  what  the  rate  per  cent.  ? 

27.  Tlie  ingredients  of  a  loaf  of  bread  weighing  Iblbs.^  are 
ie,  flour  and  water.     The  weight  of  the  rice,  augmented  by 

rlbs,^  is  two-thirds  the  weight  of  the  flour ;  and  the  weight 
of  the  water  is  onc-fifth  the  weight  of  the  flour  and  rice 
together.     Required,  the  weight  of  each. 


If 


28.  Several  detachments  of  artillery  divided  a  certain  num- 
r  of  cannon  balls.  The  first  took  72  and  J  of  the  remain- 
r;  the  next  14t  and  J  of  the  remainder;  the  third  21G 
d  J  of  the  remainder ;  the  fourth  288  and  J  of  what  was 
eft  ;  and  so  on,  until  nothing  remamed  ;  when  it  was  found 
that  the  balls  were  equally  divided.  Required,  the  number 
balls  and  the  number  of  detachments. 


29.  A  banker  has  two  kinds  of  money  ;  it  takes  a  pieceji 


300    ELEMENTARY    ALGSBRA S  U  P  P  L  E  M  E  N  "if  , 

of  the  first  to  make  a  crown,  and  h  of  the  second  to  -^^lake 
the  same  sum.  lie  is  offered  a  crown  for  c  pieces.  How 
many  of  each  kind  must  he  give  ? 

30.  Find  what  each  of  three  persons,  A,  B  and  C  is 
worth,  knowing,  1st,  that  what  A  is  worth,  added  to  I  timea 
what  B  and  C  are  worth,  is  equal  to  p ;  2d,  that  what  B  is 
worth,  added  to  m  times  what  A  and  C  are  worth,  is  equal 
to  q  ;  3d,  that  what  C  is  worth,  added  to  n  times  what  A 
and  B  are  worth,  is  equal  to  r. 

31.  Find  the  values  of  the  estates  of  six  persons,  A,  B, 
C,  D,  E  and  F,  from  the  following  conditions.  1st.  The  sum 
of  the  values  of  the  estates  of  A  and  B  is  equal  to  a  ;  that 
of  C  and  D  to  6 ;  and  that  of  E  and  F  to  c.  2d.  The 
estate  of  A  is  worth  m  times  that  of  C ;  the  estate  of  D  is 
worth  n  times  that  of  E,  and  the  estate  of  F  is  woith  p 
times  that  of  B. 


PROMISCUOUS    PROBLEMS, 

INVOLVING   EQUATIONS  OF  THE    SECOND   DEGREE, 

1.  Find  three  numbers,  such,  that  the  difference  between 
^ttie  third  and  second  shall  exceed  the  ditFerence  between  the 

I^Bcond  and  first  by  6 :  that  the  sum  of  the  numbers  shall  be 
^^,  and  the  sum  of  their  squares  467. 

2.  It  is  required  to  find  three  numbers  in  geometrical 
progression,  such  that  their  sum  shall  be  14,  and  the  sum  of 
their  squares  84. 

^^  3.  What  two  numbers  are  those,  whose  sum  multiplied 
^K  the  greater,  gives  144,  and  whose  difference  multiplied 
^^m  the  less,  gives  14 1 

^■4.  What  two  numbers  are  those,  which  are  to  each  other 
^B  m  to  7t,  and  the  sum  of  whose  squares  is  6  ] 

5.  What  two  numbers  are  those,  which  are  to  each  other 
as  771  to  n,  and  the  difference  of  whose  squares  is  b'i 

6.  A  certain  capital  is  out  at  4  per  cent,  interest.  If  we 
multiply  the  number  of  dollars  in  the  capital  by  the  num- 
ber of  dollars  in  the  interest,  for  five  months,  we  obtain 
II  17041§.     What  is  the  capital  1 

7.  A  person  has  three  kinds  of  goods,  which  together  cost 
^'iSOj^.  One  pound  of  each  article  costs  as  many  times  ^ 
i  tf  a  dollar  as  there  are  pounds  of  that  article.     Now,  he  has 


■ 


302        EJEMENTARY     ALGEBRA SUPPLEMENT. 

one-third  more  of  the  second  kind  than  of  the  first,  and  o^ 
times  more  of  the  third  than  of  the  second.  How  many 
pounds  had  he  of  each  ? 

8.  Required  to  find  three  numbers,  such,  that  the  product 
of  the  first  and  second  shall  be  equal  to  a ;  the  product  of 
the  first  and  third  equal  to  h ;  and  the  sum  of  the  squares 
of  the  second  and  third  equal  to  c. 

9.  It  is  required  to  find  three  numbers,  whose  sum  shall 
be  38,  the  sum  of  their  squares  634,  and  the  diflerence  be- 
tween the  second  and  first  greater  by  7  than  the  difference 
between  the  third  and  second. 

10.  Find  three  numbers  in  geometrical  progression,  whose 
sum  shall  be  52,  and  the  sum  of  the  extremes  to  the  mean, 
as  10  to  3. 

11.  The  sum  of  three  numbers  in  geometrical  progression 
is  13,  and  the  product  of  the  mean  by  the  sum  of  the  ex- 
tremes is  30-     What  are  the  numbers  % 

12.  It  is  required  to  find  three  numbers,  such,  that  the 
product  of  the  first  and  second,  added  to  the  sum  of  their 
squares,  shall  be  37 ;  and  the  product  of  the  first  and  third, 
added  to  the  sum  of  their  squares,  shall  be  49  ;  and  the  pro- 
duct of  the  second  and  third,  added  to  the  sum  of  theii 
squares,  shall  be  61. 

14.  Find  two  numbers,  such,  that  their  difference,  added 
to  the  difference  of  their  squares,  shall  be  equal  to  150,  and 
whose  sum,  added  to  the  sum  of  their  squares,  shall  be  equal 
to  330. 

15.  It  is  required  to  find  a  number  consisting  of  three 
digits,  such,  that  the  sum  of  the  squares  of  the  digits  shall  be 


EQUATIONS     OF     THE     SECOND     DEGREE.       .'J03 

104 ;  the  square  of  the  middle  digit  to  exceed  twice  the 
product  of  the  other  two  by  4 ;  and  if  594  be  subtracted 
from  the  number,  the  three  digits  become  inverted. 

16.  The  sum  of  two  numbers  and  the  sum  of  their  squares 
being  given,  to  find  the  numbers. 

17.  The  sum,  and  the  sum  of  the  cubes,  of  two  numbers 
being  given,  to  find  the  numbers. 

18.  To  find  three  numbers  in  arithmetical  progression 
such,  that  their  sum  shall  be  equal  to  18,  and  the  product 
of  the  two  extremes  added  to  25  shall  be  equal  to  the  square 
of  the  mean. 

19.  Having  given  the  sum,  and  the  sum  of  the  fourth 
powers  of  two  numbers  ;  to  find  the  numbers. 

20.  To  find  three  numbers  m  arithmetical  progression, 
such,  that  the  sum  of  their  squares  shall  be  equal  to  1232, 
and  the  square  of  the  mean  greater  than  the  product  of  the 
i^wo  extremes,  by  16. 

21.  To  find  two  numbers  whose  sum  is  80,  and  such,  that 
if  they  be  divided  alternately  by  each  other,  the  sum  of  the 
quotients  shall  be  3J. 

22.  To  find  two  numbers  whose  difference  shall  be  lOj 
and  if  600  be  divided  by  each  of  them,  the  difference  of  the 
quotients  shall  also  be  10. 


RETURN        Astronomy/Mathematics/Statistics  Library 
TO^^      100  Evans  Hall                        642-33E 

LOAN   PERIOD   1 

1  MO  WTO 

2                     ; 

3 

4 

5                              < 

b 

DUE  AS  STAMPED  BELOW 

"'L  1  1  200t 

FORM  NO.  DD3 


UNIVERSITY  OF  CALIFORNIA,  BERKEI 
BERKELEY,  CA  94720 


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